Methods and systems for quantum computing

ABSTRACT

The present disclosure provides methods, systems, and media for quantum computing, including allowing access to quantum ready and/or quantum enabled computers in a distributed computing environment (e.g., the cloud). Such methods and systems may provide optimization and computational services. Methods and systems of the present disclosure may enable quantum computing to be relatively and readily scaled across various types of quantum computers and users at various locations, in some cases without the need for users to have a deep understanding of the resources, implementation or the knowledge that may be required for solving optimization problems using a quantum computer. Systems provided herein may include user interfaces that enable users to perform data analysis in a distributed computing environment while taking advantage of quantum technology in the backend.

CROSS-REFERENCE

This application is a continuation-in-part of U.S. patent applicationSer. No. 15/486,960, filed Apr. 13, 2017, which is acontinuation-in-part of U.S. patent application Ser. No. 15/349,519,filed Nov. 11, 2016, now U.S. Pat. No. 9,660,859, which is acontinuation of U.S. patent application Ser. No. 15/181,247, filed Jun.13, 2016, now U.S. Pat. No. 9,537,953; U.S. patent application Ser. No.15/486,960 also claims priority to U.S. Provisional Patent ApplicationNo. 62/436,093, filed Dec. 19, 2016; this application is also acontinuation-in-part of U.S. patent application Ser. No. 15/165,655,filed May 26, 2016, each of which is entirely incorporated herein byreference.

BACKGROUND

Quantum computers typically make use of quantum-mechanical phenomena,such as superposition and entanglement, to perform operations on data.Quantum computers may be different from digital electronic computersbased on transistors. For instance, whereas digital computers requiredata to be encoded into binary digits (bits), each of which is always inone of two definite states (0 or 1), quantum computation uses quantumbits (qubits), which can be in superpositions of states.

Systems of superconducting qubits are disclosed for instance in U.S.Patent Publication No. 2012/0326720 and U.S. Publication No.2006/0225165 and manufactured by D-Wave Systems, IBM, and Google. Suchanalogue systems are used for implementing quantum computing algorithms,for example, the quantum adiabatic computation proposed by Farhi et.al., “Quantum computation by adiabatic evolution”(arXiv:quant-ph/0001106) and Grover's quantum search algorithm by L.Grover, “A fast quantum mechanical algorithm for database search”,Proceedings of the 28th Annual ACM Symposium on the Theory of Computing,pp. 212-219 (1996) and also explained in Dam et. al., “How Powerful isAdiabatic Quantum Computation?,” (arXiv:quant-ph/0206003), each of whichis entirely incorporated herein by reference.

SUMMARY

Systems and methods disclosed herein relate to quantum informationprocessing. The computational capability of a quantum computer is muchmore powerful than conventional digital computers. Quantum mechanics isnow being used to construct a new generation of computers that can solvethe most complex scientific problems—and unlock every digital vault inthe world. Such quantum computers can perform a computation in a timeperiod (e.g., seconds) that may be significantly less than a time periodof a conventional computer to perform the computation. However, the costof quantum information processing is extremely high. To make quantumcomputing more accessible to general populations, a new computationalinfrastructure integrating quantum computers and digital computers isnecessary.

Access to quantum computing resources is expensive. Therefore, a newsystem disclosed herein allows shared access to quantum computingresources. A purpose of the system disclosed herein is to providequantum computing services (e.g., optimization) on a cloud computingplatform. The quantum computing services based on today's technologieshave a potential to add additional functionalities as they aredeveloped. Using a software development kit, users are not required tohave a deep understanding of the internal architectures and mechanismsof quantum computing resources, implementation, or knowledge requiredfor solving optimization problems using a quantum computer. The systemdisclosed herein may provide user interfaces for data analysis serviceson the cloud while taking advantage of quantum technology in a backend.

Systems and methods disclosed herein may be able to improve the qualityof computing services with much greater capability, flexibility, andaffordable costs. Scalable quantum computers disclosed herein may becomplementary to digital computers wherein special-purpose computingresources are programmed or configured for certain classes of problems.Users in need of quantum computing services for their specific computingproblems can access quantum-computing resources remotely, such as on thecloud. Users can run algorithms and experiments on quantum computers andprocessors working with individual quantum bits (qubits). Users may notbe required to understand the internal architecture and mechanisms ofquantum computing resources. Users' different familiarities with theissues and relevant solutions in their respective practices, such as,for example, weather forecasting, financial analysis, cryptography,logistical planning, search for Earth-like planets, and drug discovery,etc. may provide them a flexibility of accessing different quantumcomputing resources using methods and systems disclosed herein. Quantumcomputing services provided through the cloud can provide significantlyfaster service than digital computers.

Systems and methods provided herein may improve functionality of aquantum computer, such as, for example, by providing remote access tothe quantum computer and facilitating the manner in which requests areprocessed. This can enable quantum computing to be scaled acrossmultiple users at various locations.

The present disclosure provides methods and systems that enable readyaccess to a quantum computer. Such access may be remote access or localaccess. The quantum computer may be accessed over a network, such asthrough a cloud-based interface.

The present disclosure provides systems and methods for quantuminformation processing. Many methods exist for solving a binarypolynomially constrained polynomial programming problem using a systemof superconducting qubits. The method disclosed herein can be used inconjunction with any method on any solver for solving a binarypolynomially constrained polynomial programming problem to solve amixed-integer polynomially constrained polynomial programming problem.

Current implementations of quantum devices have limited numbers ofsuperconducting qubits and are furthermore prone to various sources ofnoise. In practice, this restricts the usage of the quantum device to alimited number of qubits and a limited range of applicable local fieldbiases and couplings strengths. Therefore there is need for methods ofefficient encoding of data on the qubits of a quantum device.

Disclosed invention herein relates to quantum information processing.This application pertains to a method for storing integers onsuperconducting qubits and setting a system of such superconductingqubits having a Hamiltonian representative of a polynomial on a boundedinteger domain.

The method disclosed herein can be used as a preprocessing step forsolving a mixed integer polynomially constrained polynomial programmingproblem with a solver for binary polynomially constrained polynomialprogramming problems. One way to achieve the mentioned conversion is tocast each integer variable x as a linear function of binary variables,y_(i) for i=1, . . . , d:x=Σ _(i=1) ^(d) c _(i) y _(i).The tuple (c₁, . . . , c_(d)) is what's referred to as an integerencoding. A few well-known integer encodings are:

Binary Encoding, in which c_(i)=2^(i-1).

Unary Encoding, in which c_(i)=1.

Sequential Encoding, in which c_(i)=i.

Current implementations of quantum devices have limited numbers ofsuperconducting qubits and are furthermore prone to various sources ofnoise, including thermal and decoherence effects of the environment andthe system as disclosed by Katzgraber et. al., “Seeking quantum speedupthrough spin glasses: the good, the bad, and the ugly”(arXiv:1505.01545v2). In practice, this restricts the usage of thequantum device to a limited number of qubits and a limited range ofapplicable ferromagnetic biases and couplings.

Consequently the integer encodings formulated above, become incompetentfor representing polynomial in several integer variables as theHamiltonian of the systems mentioned above. The unary encoding suffersfrom exploiting a large number of qubits and on the other hand, in thebinary and sequential encoding the coefficients c_(i) can be too largeand therefore the behavior of the system is affected considerably by thenoise.

In an aspect, disclosed herein is a method for setting a system ofsuperconducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain via bounded-coefficient encoding,the method comprising: using one or more computer processors to obtain(i) the polynomial on the bounded integer domain and (ii) integerencoding parameters; computing a bounded-coefficient encoding using theinteger encoding parameters; recasting each integer variable as a linearfunction of binary variables using the bounded-coefficient encoding, andproviding additional constraints on the attained binary variables toavoid degeneracy in the encoding, if required by a user; substitutingeach integer variable with an equivalent binary representation, andcomputing the coefficients of the equivalent binary representation ofthe polynomial on the bounded integer domain; performing a degreereduction on the obtained equivalent binary representation of thepolynomial on the bounded integer domain to provide an equivalentpolynomial of degree at most two in binary variables; and setting localfield biases and coupling strengths on the system of superconductingqubits using the coefficients of the derived polynomial of degree atmost two in several binary variables. In some embodiments, thepolynomial on a bounded integer domain is a single bounded integervariable. In further embodiments, setting local field biases andcoupling strengths comprises assigning a plurality of qubits to have aplurality of corresponding local field biases; each local field biascorresponding to each of the qubits in the plurality of qubits isprovided using the parameters of the integer encoding. In someembodiments, the polynomial on a bounded integer domain is a linearfunction of several bounded integer variables. In further embodiments,setting local field biases and coupling strengths comprises assigning aplurality of qubits to have a plurality of corresponding local fieldbiases; each local field bias corresponding to each of the qubits in theplurality of qubits is provided using the linear function and parametersof the integer encoding. In some embodiments, the polynomial on abounded integer domain is a quadratic polynomial of several boundedinteger variables. In further embodiments, setting local field biasesand coupling strengths comprises embedding the equivalent binaryrepresentation of the polynomial of degree at most two on a boundedinteger domain to the layout of a system of superconducting qubitscomprising local fields on each of the plurality of the superconductingqubits and couplings in a plurality of pairs of the plurality of thesuperconducting qubits. In some embodiments, the system ofsuperconducting qubits is a quantum annealer. In further embodiments,the method comprises performing an optimization of the polynomial on abounded integer domain via bounded-coefficient encoding. In furtherembodiments, the optimization of the polynomial on a bounded integerdomain via bounded-coefficient encoding is obtained by quantum adiabaticevolution of an initial transverse field on the superconducting qubitsto the final Hamiltonian on a measurable axis. In further embodiments,the optimization of the polynomial on a bounded integer domain viabounded-coefficient encoding comprises: providing the equivalentpolynomial of degree at most two in binary variables; providing a systemof non-degeneracy constraints; and solving the problem of optimizationof the equivalent polynomial of degree at most two in binary variablessubject to the system of non-degeneracy constraints as a binarypolynomially constrained polynomial programming problem. In someembodiments, the method comprises solving a polynomially constrainedpolynomial programming problem on a bounded integer domain viabounded-coefficient encoding. In some embodiments, solving thepolynomially constrained polynomial programming problem on a boundedinteger domain via bounded-coefficient encoding is obtained by quantumadiabatic evolution of an initial transverse field on thesuperconducting qubits to the final Hamiltonian on a measurable axis. Infurther embodiments, solving the polynomially constrained polynomialprogramming problem on a bounded integer domain via bounded-coefficientencoding comprises: computing the bounded-coefficient encoding of theobjective function and constraints of the polynomially constrainedpolynomial programming problem using the integer encoding parameters toobtain an equivalent polynomially constrained polynomial programmingproblem in several binary variables; providing a system ofnon-degeneracy constraints; adding the system of non-degeneracyconstraints to the constraints of the obtained polynomially constrainedpolynomial programming problem in several binary variables; and solvingthe problem of optimization of the obtained polynomially constrainedpolynomial programming problem in several binary variables. In someembodiments, the obtaining of integer encoding parameters comprisesobtaining an upper bound on the coefficients of the bounded-coefficientencoding directly. In some embodiments, the obtaining of integerencoding parameters comprises obtaining an upper bound on thecoefficients of the bounded-coefficient encoding based on errortolerances ∈_(l) and ∈_(c) of local field biases and couplings strengthsof the system of superconducting qubits. In some embodiments, obtainingan upper bound on the coefficient of the bounded-coefficient encodingcomprises finding a feasible solution to a system of inequalityconstraints.

In another aspect, disclosed herein is a system comprising: a sub-systemof superconducting qubits; a computer operatively coupled to thesub-system of superconducting qubits, wherein the computer comprises atleast one computer processor, an operating system configured to performexecutable instructions, and a memory; and a computer program includinginstructions executable by the at least one computer processor togenerate an application for setting the sub-system of superconductingqubits having a Hamiltonian representative of a polynomial on a boundedinteger domain via bounded-coefficient encoding, the applicationcomprising: a software module programmed or otherwise configured toobtain the polynomial on the bounded integer domain; a software moduleprogrammed or otherwise configured to obtain integer encodingparameters; a software module programmed or otherwise configured tocompute a bounded-coefficient encoding using the integer encodingparameters; a software module programmed or otherwise configured torecast each integer variable as a linear function of binary variablesusing the bounded-coefficient encoding, and providing additionalconstraints on the attained binary variables to avoid degeneracy in theencoding, if required by a user; a software module programmed orotherwise configured to substitute each integer variable with anequivalent binary representation, and compute the coefficients of theequivalent binary representation of the polynomial on the boundedinteger domain; a software module programmed or otherwise configured toperform a degree reduction on the obtained equivalent binaryrepresentation of the polynomial on the bounded integer domain toprovide an equivalent polynomial of degree at most two in binaryvariables; and a software module programmed or otherwise configured toset local field biases and coupling strengths on the system ofsuperconducting qubits using the coefficients of the derived polynomialof degree at most two in several binary variables. In some embodiments,the polynomial on a bounded integer domain is a single bounded integervariable. In further embodiments, setting local field biases andcoupling strengths comprises assigning a plurality of qubits to have aplurality of corresponding local field biases; each local field biascorresponding to each of the qubits in the plurality of qubits isprovided using the parameters of the integer encoding. In someembodiments, the polynomial on a bounded integer domain is a linearfunction of several bounded integer variables. In further embodiments,setting local field biases and coupling strengths comprises assigning aplurality of qubits to have a plurality of corresponding local fieldbiases; each local field bias corresponding to each of the qubits in theplurality of qubits is provided using the linear function and parametersof the integer encoding. In some embodiments, the polynomial on abounded integer domain is a quadratic polynomial of several boundedinteger variables. In further embodiments, setting local field biasesand coupling strengths comprises embedding the equivalent binaryrepresentation of the polynomial of degree at most two on a boundedinteger domain to the layout of a system of superconducting qubitscomprising local fields on each of the plurality of the superconductingqubits and couplings in a plurality of pairs of the plurality of thesuperconducting qubits. In some embodiments, the system ofsuperconducting qubits is a quantum annealer. In further embodiments,the system comprises performing an optimization of the polynomial on abounded integer domain via bounded-coefficient encoding. In furtherembodiments, the optimization of the polynomial on a bounded integerdomain via bounded-coefficient encoding is obtained by quantum adiabaticevolution of an initial transverse field on the superconducting qubitsto the final Hamiltonian on a measurable axis. In further embodiments,the optimization of the polynomial on a bounded integer domain viabounded-coefficient encoding comprises: providing the equivalentpolynomial of degree at most two in binary variables; providing a systemof non-degeneracy constraints; and solving the problem of optimizationof the equivalent polynomial of degree at most two in binary variablessubject to the system of non-degeneracy constraints as a binarypolynomially constrained polynomial programming problem. In someembodiments, the system comprises solving a polynomially constrainedpolynomial programming problem on a bounded integer domain viabounded-coefficient encoding. In some embodiments, solving thepolynomially constrained polynomial programming problem on a boundedinteger domain via bounded-coefficient encoding is obtained by quantumadiabatic evolution of an initial transverse field on thesuperconducting qubits to the final Hamiltonian on a measurable axis. Infurther embodiments, solving the polynomially constrained polynomialprogramming problem on a bounded integer domain via bounded-coefficientencoding comprises: computing the bounded-coefficient encoding of theobjective function and constraints of the polynomially constrainedpolynomial programming problem using the integer encoding parameters toobtain an equivalent polynomially constrained polynomial programmingproblem in several binary variables; providing a system ofnon-degeneracy constraints; adding the system of non-degeneracyconstraints to the constraints of the obtained polynomially constrainedpolynomial programming problem in several binary variables; and solvingthe problem of optimization of the obtained polynomially constrainedpolynomial programming problem in several binary variables. In someembodiments, the obtaining of integer encoding parameters comprisesobtaining an upper bound on the coefficients of the bounded-coefficientencoding directly. In some embodiments, the obtaining of integerencoding parameters comprises obtaining an upper bound on thecoefficients of the bounded-coefficient encoding based on errortolerances ∈_(l) and ∈_(c) of local field biases and couplings strengthsof the system of superconducting qubits. In some embodiments, obtainingan upper bound on the coefficient of the bounded-coefficient encodingcomprises finding a feasible solution to a system of inequalityconstraints.

In another aspect, disclosed herein is a non-transitorycomputer-readable medium comprising machine-executable code that, uponexecution by one or more computer processors, implements a method forsetting a system of superconducting qubits having a Hamiltonianrepresentative of a polynomial on a bounded integer domain viabounded-coefficient encoding, the method comprising: using one or morecomputer processors to obtain (i) the polynomial on the bounded integerdomain and (ii) integer encoding parameters; computing thebounded-coefficient encoding using the integer encoding parameters;recasting each integer variable as a linear function of binary variablesusing the bounded-coefficient encoding, and providing additionalconstraints on the attained binary variables to avoid degeneracy in theencoding, if required by a user; substituting each integer variable withan equivalent binary representation, and computing the coefficients ofthe equivalent binary representation of the polynomial on the boundedinteger domain; performing a degree reduction on the obtained equivalentbinary representation of the polynomial on the bounded integer domain toprovide an equivalent polynomial of degree at most two in binaryvariables; and setting local field biases and coupling strengths on thesystem of superconducting qubits using the coefficients of the derivedpolynomial of degree at most two in several binary variables.

Disclosed is a method for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding, the method comprising obtaining(i) the polynomial on the bounded integer domain and (ii) integerencoding parameters; computing the bounded-coefficient encoding usingthe integer encoding parameters; recasting each integer variable as alinear function of binary variables using the bounded-coefficientencoding, and providing additional constraints on the attained binaryvariables to avoid degeneracy in the encoding, if required by a user;substituting each integer variable with an equivalent binaryrepresentation, and computing the coefficients of the equivalent binaryrepresentation of the polynomial on the bounded integer domain;performing a degree reduction on the obtained equivalent binaryrepresentation of the polynomial on the bounded integer domain toprovide an equivalent polynomial of degree at most two in binaryvariables; and setting local field biases and coupling strengths on thesystem of superconducting qubits using the coefficients of the derivedpolynomial of degree at most two in several binary variables.

In some embodiments, the obtaining of a polynomial in n variables on abounded integer domain comprises of providing the plurality of terms inthe polynomial; each term of the polynomial further comprises of thecoefficient of the term and a list of size n representative of the powerof each variables in the term in the matching index. The obtaining of apolynomial on a bounded integer domain further comprises of obtaining alist of upper bounds on each integer variable.

In a particular case where the provided polynomial is of degree at mosttwo, the obtaining of a polynomial on bounded domain comprises ofproviding coefficients q_(i) of each linear term x_(i) for i=1, . . . ,n, and coefficients Q_(ij)+Q_(ji) of each quadratic term x_(i)x_(j) forall choices of distinct elements {i,j}⊆{1, . . . , n} and an upper boundon each integer variable.

In some embodiments, the obtaining of integer encoding parameterscomprises of either obtaining an upper bound on the value of thecoefficients of the encoding directly; or obtaining the error tolerance∈_(l) and ∈_(c) of the local field biases and couplings, respectively,and computing the upper bound of the coefficients of the encoding fromthese error tolerances. This application proposes a technique forcomputing upper bound of the coefficients of the encoding from ∈_(l) and∈_(c) for the special case that the provided polynomial is of degree atmost two.

In some embodiments, the integer encoding parameters are obtained fromat least one of a user, a computer, a software package and anintelligent agent.

In some embodiments, the bounded-coefficient encoding is derived and theinteger variables are represented as a linear function of a set ofbinary variables using the bounded-coefficient encoding, and a system ofnon-degeneracy constraints is returned.

In another aspect, disclosed is a digital computer comprising: a centralprocessing unit; a display device; a memory unit comprising anapplication for storing data and computing arithmetic operations; and adata bus for interconnecting the central processing unit, the displaydevice, and the memory unit.

In another aspect, there is disclosed a non-transitory computer-readablestorage medium for storing computer-executable instructions which, whenexecuted, cause a digital computer to perform arithmetic and logicaloperations.

In another aspect, there is disclosed a system of superconducting qubitscomprising; a plurality of superconducting qubits; a plurality ofcouplings between a plurality of pairs of superconducting qubits; aquantum device control system capable of setting local field biases oneach of the superconducting qubits and couplings strengths on each ofthe couplings.

The method disclosed herein makes it possible to represent a polynomialon a bounded integer domain on a system of superconducting qubits. Themethod comprises of obtaining (i) the polynomial on the bounded integerdomain and (ii) integer encoding parameters; computing thebounded-coefficient encoding using the integer encoding parameters;recasting each integer variable as a linear function of binary variablesusing the bounded-coefficient encoding, and providing additionalconstraints on the attained binary variables to avoid degeneracy in theencoding, if required by a user; substituting each integer variable withan equivalent binary representation, and computing the coefficients ofthe equivalent binary representation of the polynomial on the boundedinteger domain; performing a degree reduction on the obtained equivalentbinary representation of the polynomial on the bounded integer domain toprovide an equivalent polynomial of degree at most two in binaryvariables; and setting local field biases and coupling strengths on thesystem of superconducting qubits using the coefficients of the derivedpolynomial of degree at most two in several binary variables.

In some embodiments of this application, the method disclosed hereinmakes it possible to find the optimal solution of a mixed integerpolynomially constrained polynomial programming problem through solvingits equivalent binary polynomially constrained polynomial programmingproblem. In one embodiment, solving a mixed integer polynomiallyconstrained polynomial programming problem comprises finding a binaryrepresentation of all polynomials appearing the objective function andthe constraints of the problem using the bounded-coefficient encodingand applying the methods disclosed in U.S. patent application Ser. No.15/051,271, U.S. patent application Ser. No. 15/014,576, CA PatentApplication No. 2921711 and CA Patent Application No. 2881033, each ofwhich is entirely incorporated herein by reference, to the obtainedequivalent binary polynomially constrained polynomial programmingproblem.

In yet another aspect, the present disclosure provides a method forusing a digital computer to generate and direct a computational task toa quantum computing resource comprising at least one quantum computerover a network, wherein the digital computer comprises at least onecomputer processor and at least one computer memory, the methodcomprising: retrieving a programming problem from the computer memory ofthe digital computer; using the at least one computer processor of thedigital computer to generate an equivalent of the programming problem;generating a request comprising the equivalent of the programmingproblem generated by the at least one computer processor of the digitalcomputer; and directing the request from the digital computer to thequantum computing resource over the network, wherein the equivalent ofthe programming problem is usable by the at least one quantum computerof the quantum computing resource to solve the programming problem.

In some embodiments, the request is directed from the digital computerto the quantum computing resource through a cloud-based interface. Insome embodiments, the network is a local network. In some embodiments,the at least one quantum computer performs one or more quantumalgorithms to the programming problem.

In some embodiments, the request is generated using an applicationprogramming interface (API).

In some embodiments, wherein the method further comprises obtaining (i)a polynomial on a bounded integer domain and (ii) integer encodingparameters, and computing a bounded-coefficient encoding using theinteger encoding parameters. In some embodiments, the method furthercomprises using the one or more computer processors to transform eachinteger variable of the polynomial to a linear function of binaryvariables using the bounded-coefficient encoding. In some embodiments,the method further comprises providing constraints on the binaryvariables to avoid degeneracy in the bounded-coefficient encoding, ifrequired by a user. In some embodiments, the method further comprisessubstituting each integer variable of the polynomial with an equivalentbinary representation, and using the at least one computer processor tocompute coefficients of an equivalent binary representation of thepolynomial on the bounded integer domain. In some embodiments, themethod further comprises performing a degree reduction on the equivalentbinary representation of the polynomial on the bounded integer domain togenerate an equivalent polynomial. In some embodiments, the equivalentpolynomial is of a degree of at most two in binary variables. In someembodiments, the method further comprises setting local field biases andcoupling strengths on the at least one quantum computer using thecoefficients of the equivalent polynomial of the degree of at most twoin binary variables to generate the equivalent of the programmingproblem. In some embodiments, the equivalent of the programming problemcomprises a Hamiltonian representative of the polynomial on the boundedinteger domain. In some embodiments, the Hamiltonian is usable by the atleast one quantum computer to solve the programming problem.

In yet another aspect, the present disclosure provides a systemcomprising a digital computer for generating and directing acomputational task to a quantum computing resource comprising at leastone quantum computer over a network, wherein the digital computercomprises at least one computer processor and at least one computermemory, wherein the at least one computer processor is programmed to:retrieve a programming problem from the computer memory of the digitalcomputer; use the at least one computer processor of the digitalcomputer to generate an equivalent of the programming problem; generatea request comprising the equivalent of the programming problem generatedby the at least one computer processor of the digital computer; anddirect the request from the digital computer to the quantum computingresource over the network, wherein the equivalent of the programmingproblem is usable by the at least one quantum computer of the quantumcomputing resource to solve the programming problem.

In some embodiments, the at least one computer processor is programmedto direct the request from the digital computer to the quantum computingresource through a cloud-based interface. In some embodiments, thenetwork is a local network.

In some embodiments, the at least one computer processor is programmedto obtain (i) a polynomial on a bounded integer domain and (ii) integerencoding parameters, and compute a bounded-coefficient encoding usingthe integer encoding parameters. In some embodiments, the at least onecomputer processor is programmed to (i) transform each integer variableof the polynomial to a linear function of binary variables using thebounded-coefficient encoding, and (ii) substitute each integer variableof the polynomial with an equivalent binary representation, and usingthe at least one computer processor to compute coefficients of anequivalent binary representation of the polynomial on the boundedinteger domain. In some embodiments, the at least one computer processoris programmed to perform a degree reduction on the equivalent binaryrepresentation of the polynomial on the bounded integer domain togenerate an equivalent polynomial, wherein the equivalent polynomial isof a degree of at most two in binary variables. In some embodiments, theat least one computer processor is programmed to set local field biasesand coupling strengths on the at least one quantum computer using thecoefficients of the equivalent polynomial of the degree of at most twoin binary variables to generate the equivalent of the programmingproblem. In some embodiments, the equivalent of the programming problemcomprises a Hamiltonian representative of the polynomial on the boundedinteger domain. In some embodiments, the Hamiltonian is usable by the atleast one quantum computer to solve the programming problem.

Additional aspects and advantages of the present disclosure will becomereadily apparent to those skilled in this art from the followingdetailed description, wherein only illustrative embodiments of thepresent disclosure are shown and described. As will be realized, thepresent disclosure is capable of other and different embodiments, andits several details are capable of modifications in various obviousrespects, all without departing from the disclosure. Accordingly, thedrawings and description are to be regarded as illustrative in nature,and not as restrictive.

INCORPORATION BY REFERENCE

All publications, patents, and patent applications mentioned in thisspecification are herein incorporated by reference to the same extent asif each individual publication, patent, or patent application wasspecifically and individually indicated to be incorporated by reference.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features of the invention are set forth with particularity inthe appended claims. A better understanding of the features andadvantages of the present invention will be obtained by reference to thefollowing detailed description that sets forth illustrative embodiments,in which the principles of the invention are utilized, and theaccompanying drawings (also “figure” and “FIG.” herein), of which:

FIG. 1 shows a non-limiting example of an Application Program Interface(API) gateway and a queuing unit.

FIG. 2 shows a non-limiting example of an API gateway, a queuing unit,and a database service.

FIG. 3 shows a non-limiting example of a queuing unit, database service,and a cluster manager.

FIG. 4 shows a non-limiting example of a cluster manager and a loggingunit.

FIG. 5 shows a non-limiting example of a computing architecture of acloud platform for accessing shared quantum computing resources.

FIG. 6 shows a non-limiting example of a quantum-enabled computingplatform.

FIG. 7 shows a non-limiting example of an analysis tree for decomposinga given problem into sub-problems in quantum and classical computingresources.

FIG. 8 shows a non-limiting example of a method for setting a system ofsuperconducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain; in this case, a flowchart of allsteps used for setting a system of superconducting qubits in such a way.

FIG. 9 shows a non-limiting example of a method for setting a system ofsuperconducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain; in this case, a diagram of asystem comprising of a digital computer interacting with a system ofsuperconducting qubits.

FIG. 10 shows a non-limiting example of a method for setting a system ofsuperconducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain; in this case, a detailed diagramof a system comprising of a digital computer interacting with a systemof superconducting qubits used for computing the local fields andcouplers.

FIG. 11 shows a non-limiting example of a method for setting a system ofsuperconducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain; in this case, a flowchart of astep for providing a polynomial on a bounded integer domain.

FIG. 12 shows a non-limiting example of a method for setting a system ofsuperconducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain; in this case, a flowchart of astep for providing encoding parameters.

FIG. 13 shows a non-limiting example of a method for setting a system ofsuperconducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain; in this case, a flowchart of astep for computing the bounded-coefficient encoding.

FIG. 14 shows a non-limiting example of a method for setting a system ofsuperconducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain; in this case, a flowchart of astep for converting a polynomial on a bounded integer domain to anequivalent polynomial in several binary variables.

DETAILED DESCRIPTION

While various embodiments of the invention have been shown and describedherein, it will be obvious to those skilled in the art that suchembodiments are provided by way of example only. Numerous variations,changes, and substitutions may occur to those skilled in the art withoutdeparting from the invention. It should be understood that variousalternatives to the embodiments of the invention described herein may beemployed.

Methods and Systems for Non-Classical Computing on the Cloud

Quantum computing resources may be rare. Access to quantum computingresources may be expensive or such quantum computing resources may beinaccessible given geographic limitations. Even though a user may havedirect access to a quantum computer, the user may be required to possesssophisticated expertise to configure the quantum computer and/or choosean adequate quantum algorithm for solving a computational task;otherwise, the user does not gain the benefit from the speedycomputations offered by the quantum computer. Even a superior quantumcomputer may not exhibit any advantage over classical computingresources in solving a problem if the right algorithm, the rightproblem, and the right parameters are not chosen. On the other hand,from a user's perspective, a computational problem may be a very largecomputational task involving many smaller sub-tasks. Each of thesesub-tasks may possess a different complexity characteristic. Therefore,using the right computing resource, the right algorithm, and the rightparameter may be essential to solve the original problem efficientlyand/or benefit from the potential quantum speedup.

The present disclosure provides systems and methods that offerquantum-ready services and/or quantum-enabled services. Quantum-readyservices may advantageously make it easier for a user to manage quantumresources and switch between a classical or quantum computationresource. Additionally, a quantum-enabled framework may allow users touse both classical and quantum resources in a hybrid manner such thatthe framework intelligently chooses the right solver and the rightparameters for each particular sub-problem or subtask.

The present disclosure provides systems and methods that may allowshared or distributed access to quantum computing resources (e.g., aquantum-ready or quantum-enabled services). The disclosed system mayprovide quantum computing services (e.g., optimization based on quantumalgorithms) on a cloud computing platform. Using a software developmentkit (SDK), users may not be required to have a deep understanding of thequantum computing resources, implementation, or the knowledge requiredfor solving optimization problems using a quantum computer. For example,use of an SDK to provide a user with shared or distributed access toquantum computing resources is disclosed in PCT InternationalApplication PCT/CA2017/050320, “Methods and Systems for QuantumComputing,” which is entirely incorporated herein by reference.

The present disclosure provides systems and methods for facilitatingquantum computing in a distributed environment, such as over a network(e.g., in the cloud). For example, a user at a first location may submita request for a calculation or task to be performed by a quantumcomputer (e.g., an adiabatic quantum computer) at a second location thatis remotely located with respect to the first location. The request maybe directed over a network to one or more computer servers, whichsubsequently direct a request to the quantum computer to perform thecalculation or task.

Provided herein are systems and methods that provide optimizationservices in a distributed computing environment (e.g., the cloud), whichmay utilize quantum computing technology, such as an adiabatic quantumcomputer. Methods and systems of the present disclosure enable quantumcomputing to be relatively and readily scaled across various types ofquantum computers and users in various locations, in some cases withouta need for users to have a deep understanding of the resources,implementation, or the knowledge required for solving optimizationproblems using a quantum computer. Systems provided herein may includeuser interfaces that enable users to perform data analysis in adistributed computing environment (e.g., in the cloud) while takingadvantage of quantum technology in the backend.

In some embodiments, systems, media, networks, and methods include aquantum computer, or use of the same. Quantum computation uses quantumbits (qubits), which can be in superpositions of states. A quantumTuring machine is a theoretical model of such a computer, and is alsoknown as a universal quantum computer. Quantum computers sharetheoretical similarities with non-deterministic and probabilisticcomputers.

In some embodiments, a quantum computer comprises one or more quantumprocessors. A quantum computer may be configured to perform one or morequantum algorithms. A quantum computer may store or process datarepresented by quantum bits (qubits). A quantum computer may be able tosolve certain problems much more quickly than any classical computersthat use even the best currently available algorithms, like integerfactorization using Shor's algorithm or the simulation of quantummany-body systems. There exist quantum algorithms, such as Simon'salgorithm, that run faster than any possible probabilistic classicalalgorithm. Examples of quantum algorithms include, but are not limitedto, quantum optimization algorithms, quantum Fourier transforms,amplitude amplifications, quantum walk algorithms, and quantum evolutionalgorithms. Quantum computers may be able to efficiently solve problemsthat no classical computer may be able to solve within a reasonableamount of time. Thus, a system disclosed herein utilizes the merits ofquantum computing resources to solve complex problems.

Any type of quantum computers may be suitable for the technologiesdisclosed herein. Examples of quantum computers include, but are notlimited to, adiabatic quantum computers, quantum gate arrays, one-wayquantum computer, topological quantum computers, quantum Turingmachines, superconductor-based quantum computers, trapped ion quantumcomputers, optical lattices, quantum dot computers, spin-based quantumcomputers, spatial-based quantum computers, Loss-DiVincenzo quantumcomputers, nuclear magnetic resonance (NMR) based quantum computers,liquid-NMR quantum computers, solid state NMR Kane quantum computers,electrons-on-helium quantum computers, cavity-quantum-electrodynamicsbased quantum computers, molecular magnet quantum computers,fullerene-based quantum computers, linear optical quantum computers,diamond-based quantum computers, Bose-Einstein condensate-based quantumcomputers, transistor-based quantum computers, andrare-earth-metal-ion-doped inorganic crystal based quantum computers. Aquantum computer may comprise one or more of: a quantum annealer, anIsing solver, an optical parametric oscillator (OPO), or a gate model ofquantum computing.

A system of the present disclosure may include or employ quantum-readyor quantum-enabled computing systems. A quantum-ready computing systemmay comprise a digital computer operatively coupled to a quantumcomputer. The quantum computer may be configured to perform one or morequantum algorithms. A quantum-enabled computing system may comprise aquantum computer and a classical computer, the quantum computer and theclassical computer operatively coupled to a digital computer. Thequantum computer may be configured to perform one or more quantumalgorithms for solving a computational problem. The classical computermay comprise at least one classical processor and computer memory, andmay be configured to perform one or more classical algorithms forsolving a computational problem.

The term “quantum annealer” and like terms generally refer to a systemof superconducting qubits that carries optimization of a configurationof spins in an Ising spin model using quantum annealing, as described,for example, in Farhi, E. et al., “Quantum Adiabatic EvolutionAlgorithms versus Simulated Annealing” arXiv.org: quant ph/0201031(2002), pp. 1-16. An embodiment of such an analog processor is disclosedby McGeoch, Catherine C. and Cong Wang, (2013), “Experimental Evaluationof an Adiabatic Quantum System for Combinatorial Optimization” ComputingFrontiers,” May 14-16, 2013(http://www.cs.amherst.edu/ccm/cf14-mcgeoch.pdf) and also disclosed inU.S. Patent Application Publication Number US 2006/0225165.

In some embodiments, a classical computer may be configured to performone or more classical algorithms. A classical algorithm (or classicalcomputational task) may be an algorithm (or computational task) that isable to be executed by one or more classical computers without the useof a quantum computer, a quantum-ready computing service, or aquantum-enabled computing service. A classical algorithm may be anon-quantum algorithm. A classical computer may be a computer which doesnot comprise a quantum computer, a quantum-ready computing service, or aquantum-enabled computer. A classical computer may process or store datarepresented by digital bits (e.g., zeroes (“0”) and ones (“1”)) ratherthan quantum bits (qubits). Examples of classical computers include, butare not limited to, server computers, desktop computers, laptopcomputers, notebook computers, sub-notebook computers, netbookcomputers, netpad computers, set-top computers, media streaming devices,handheld computers, Internet appliances, mobile smartphones, tabletcomputers, personal digital assistants, video game consoles, andvehicles.

In an aspect, the present disclosure provides a system for quantum-readyoptimization. The computing system may comprise a digital computeroperatively coupled to a remote quantum computer over a network. Thequantum computer may be configured to perform one or more quantumalgorithms. The digital computer may comprise at least one computerprocessor and computer memory. The computer memory may include acomputer program with instructions executable by the at least onecomputer processor to render an application. The application mayfacilitate use of the quantum computer by a user.

In another aspect, the present disclosure provides a system forquantum-enabled optimization. The computing system may comprise aquantum computer and a classical computer, the quantum computer and theclassical computer operatively coupled to a digital computer over anetwork. The quantum computer may be configured to perform one or morequantum algorithms for solving a computational problem. The classicalcomputer may comprise at least one classical processor and computermemory, and may be configured to perform one or more classicalalgorithms for solving a computational problem. The digital computer maycomprise at least one computer processor and computer memory, whereinthe digital computer may include a computer program with instructionsexecutable by the at least one computer processor to render anapplication. The application may facilitate use of the quantum computerand/or the classical computer by a user.

Some implementations may use quantum computers along with classicalcomputers operating on bits, such as personal desktops, laptops,supercomputers, distributed computing, clusters, cloud-based computingresources, smartphones, or tablets.

The system may include a gateway programmed or configured to receive arequest over the network. The request may comprise a computational task.Examples of a computational task include, but are not limited to,search, optimization, statistical analysis, modeling, data processing,etc. In some embodiments, a request may comprise a dataset; for example,a data matrix including variables and observations for creating amodeling or analyzing statistics of the data set. Further, a solutionmay be derived; for example, an optimal model underlying a given datasetis derived from a quantum computer; a statistical analysis is performedby a quantum computer.

The system may comprise a queuing unit programmed or configured to storeand order the request in one or more queues. The system may comprise acluster manager programmed or configured to create an instance/container(also “worker” herein) to (1) translate the request in the queue intoone or more quantum machine instructions, (2) deliver the one or morequantum machine instructions to the quantum computer over the network toperform the computational task, and (3) receive one or more solutionsfrom the quantum computer. The one or more solutions may be stored in adatabase of the system. The system may comprise a logging unitprogrammed or configured to log an event of the worker.

The system may comprise an interface for a user. In some embodiments,the interface may comprise an application programming interface (API).The interface may provide a programmatic model that abstracts away(e.g., by hiding from the user) the internal details (e.g., architectureand operations) of the quantum computer. In some embodiments, theinterface may minimize a need to update the application programs inresponse to changing quantum hardware. In some embodiments, theinterface may remain unchanged when the quantum computer has a change ininternal structure.

Gateway

Systems, media, networks, and methods of the present disclosure maycomprise a gateway that may be programmed or configured to receive arequest from a user. The request may comprise a computational task. Insome embodiments, the gateway is programmed or configured toauthenticate a user of the system. In some embodiments, the gateway isprogrammed or configured to monitor system and data security. As anexample, a gateway may use secure sockets layer (SSL) for encryptingrequests and responses. In some embodiments, a gateway is programmed orconfigured to route the request to one of the at least one digitalprocessor. In some embodiments, a gateway is programmed or configured tomonitor data traffic.

In some embodiments, the systems, media, networks, and methods comprisea queuing unit. In some embodiments, a queuing unit is programmed orconfigured to place the request in the queue. When a queue comprisesmore than one request, the more than one requests may be placed inorder. The order may be based on first-in-first-out, or based on timing,or based on available quantum computing resources. In some embodiments,a queuing unit is further programmed or configured to reorder therequest in the queue. In some embodiments, a queuing unit is responsiblefor preventing message loss. The tasks submitted may be stored in thequeue and may be accessed in order by the microservices that need towork with them.

A gateway may be a microservice used for authentication, routing,security, and monitoring purposes. Referring to FIG. 1, a request 101 isreceived by an application programming interface (API) gateway 111 andthen forwarded through to one or more target microservices. In someembodiments, when the target microservices are not availableimmediately, the request 101 may be first handled by a queuing unit 121which places the request in a queue. In some cases, a request is pushedinto the queue 121 or inserted into the queue 121. In some embodiments,a request in the queue 121 is reordered based on priorities ofcomputational tasks. For instance, if a new incoming request has a samecomputational task as the request at the top of the queue, to savequantum computing resources, it may be better to have the new requestbeing executed concurrently with the top queue, so the queuing unitplaces the new request at the top of the queue as well.

In some embodiments, the systems, media, networks, and methods describedherein comprise a database service, or use of the same. In someembodiments, a database is programmed or configured to store a data setin the request. In some embodiments, a database in the microservices isin charge of storing persistent data. In some embodiments, solutions tosolved problems are maintained by the database.

Referring to FIG. 2, a database 131 communicates with the queuing unit121. In some embodiments, status of a worker or a quantum computingresource (e.g., availabilities, reading, writing, queuing, algorithms tobe executed, algorithms having been performed, and timestamps) arestored in the database 131. In some embodiments, data sent along with arequest are stored in the database 131 as well. Persistent data andsolutions to solved tasks may be stored in the database 131.

In some embodiments, the quantum-ready system disclosed herein comprisesone or more databases, or use of the same. Many types of databases maybe suitable for storage and retrieval of application information. Insome embodiments, suitable databases include, by way of non-limitingexamples, relational databases, non-relational databases, objectoriented databases, object databases, entity-relationship modeldatabases, associative databases, and XML databases. In someembodiments, a database is internet-based. In some embodiments, adatabase is web-based. In some embodiments, a database is cloudcomputing-based (e.g., on the cloud). In other embodiments, a databaseis based on one or more local computer storage devices.

In some embodiments, a system may comprise a serialization unitconfigured to communicate problem instances from the user to the quantumcomputer through the gateway. On the other hand, the serialization unitmay be programmed or configured to communicate computed solutions tothose instances from the quantum computer back to the user through thegateway. The serialization mechanism may be based on JavaScript ObjectNotation (JSON), eXtensible Markup Language (XML), or other markuplanguages; however, an entirely new format for the serialization may beused. In some embodiments, the serialization mechanism may comprisetransmitting texts or binary files. In some embodiments, theserialization mechanism may or may not be encrypted. In someembodiments, the serialization mechanism may or may not be compressed.

In some embodiments, a system may comprise a user interface configuredto allow a user to submit a request to solve a computational task. Auser may specify the task and submit associated datasets. The userinterface may transmit the request and the datasets to the gateway. Thegateway may then process the request based on the technologies disclosedherein. When solutions are derived by a quantum computer, the gatewaymay send a notification to the user. The user may retrieve the solutionsvia the user interface.

Cluster Manager

In some embodiments, the systems, media, networks, and methods describedherein comprise a cluster manager. The cluster manager may be programmedor configured to translate the request into quantum machineinstructions. In some embodiments, the cluster manager delivers thequantum machine instructions to a quantum processor to perform thecomputational task. In addition, the cluster manager receives one ormore solutions from the quantum processor.

In some embodiments, a cluster manager is programmed to divide thecomputational task into two or more computational components. In someembodiments, a computational component corresponds to a quantumalgorithm. In some embodiments, the two or more computational componentsare translated into one or more quantum algorithms, or translated intoquantum machine instructions.

In some embodiments, translating into quantum machine instructionscomprises determination of a number of qubits and/or determination of aquantum operator. In some embodiments, two or more computationalcomponents are executed by the quantum computer sequentially, inparallel, or both thereof.

In some embodiments, a cluster manager is programmed or configured toaggregate solutions of the two or more computational components. In someembodiments, a cluster manager is further programmed or configured tocontrol a start and a termination of the computational task. Further, acluster manager may be programmed or configured to monitor a lifetime ofthe computational task.

Referring to FIG. 3, the central queue 121 transmits the recent state ofthe queue to the cluster manager 141. In this example, the clustermanager 141 is realized by an Apache Mesos server. The cluster manager141 starts and controls a lifetime of certain types of computationalcomponents. The cluster manager starts instances/containers (calledworkers) that are able to perform the operations (e.g., translating tospecific quantum computing instructions, controlling quantumcomputers/processors 201 to execute computational tasks, etc.) requiredby the queue entries. For instance, a worker 152 is assigned to processa request 122. If the worker 152 is successful, it sends a result of theprocessing of the request to the database service 131 and removes theentry 122 from the queue. The worker 152 is then destroyed to free upresources for other operations in order to save costs.

In some embodiments, an algorithm specified in a request may comprise aclassical or a quantum algorithm. A worker may determine if theclassical algorithm or the quantum algorithm has to be translated intoanother classical algorithm or another quantum algorithm. Once acomputational task in a request has been translated into quantum machineinstructions, the quantum machine instructions may be transmitted to aquantum computer. The quantum computer may execute a classical algorithmor a quantum algorithm or both to complete a computational task.

Logging Unit

In some embodiments, the systems, media, networks, and methods describedherein comprise a logging unit, or use of the same. In some embodiments,a logging service is in charge of tracking the events occurring inseparate microservices. Some or all of the microservices may transmit alog of events into a central logging microservice.

An event disclosed herein may be associated with any one or more of thefollowing: a login into the system, submitting a request, processing therequest, queuing the request, processing a computational task in therequest, dividing the computational task, translating the computationaltask into a quantum algorithm and quantum instructions, transmittingquantum instructions to a quantum computer, performing computations by aquantum computer, performing computational operations in a quantumcomputer, transmitting a computational result or a solution from aquantum computer to a server, and notifying a user of an availability ofthe results or solutions.

In some embodiments, a logging unit is programmed or configured to storea log, wherein the log comprises an event taking in the digital computeror the quantum processor. In some embodiments, a log comprises atimestamp of the event.

Referring to FIG. 4, a logging unit 161 communicates with workers torecord some or all of the events. In this figure, the logging unit 161communicates with a worker 153 to record the start, the operations, andthe end of computational tasks.

Quantum-Enabled and Quantum-Ready Computing

The present disclosure provides systems, media, networks, and methodsthat may include quantum-enabled computing or use of quantum-enabledcomputing. Quantum computers may be able to solve certain classes ofcomputational tasks more efficiently than classical computers. However,quantum computation resources may be rare and expensive, and may involvea certain level of expertise to be used efficiently or effectively(e.g., cost-efficiently or cost-effectively). A number of parameters maybe tuned in order for a quantum computer to deliver its potentialcomputational power.

Quantum computers (or other types of non-classical computers) may beable to work alongside classical computers as co-processors. A hybridarchitecture of quantum-enabled computation can be very efficient foraddressing complex computational tasks, such as hard optimizationproblems. A system disclosed herein may provide a remote interfacecapable of solving computationally expensive problems by deciding if aproblem may be solved efficiently on a quantum-ready or a classicalcomputing service. The computing service behind the interface may beable to efficiently and intelligently decompose or break down theproblem and delegate appropriate components of the computational task toa quantum-ready or a classical service.

The methods and systems described here may comprise an architectureconfigured to realize a cloud-based framework to provide hybridquantum-enabled computing solutions to complex computational problems(such as complex discrete optimization) using a classical computer forsome portion of the work and a quantum (or quantum-like) computer (e.g.,quantum-ready or quantum-enabled) for the remaining portion of the work.

FIG. 6 shows a workflow for performing a computational task using aquantum-ready (quantum ready) service. In a first operation, a user or aclient may submit a computational task to an API gateway 601. Thecomputational task may then be forwarded to an arbiter 602. In a secondoperation, the arbiter 602 may decompose and/or distribute thecomputational task to a quantum ready service 603 and a classicalservice 604. The computational task may thus be decomposed and/ordistributed into sub-problems, each of which may be performed by arespective one of the quantum ready service 603 and the classicalservice 604. Next, one or more solutions from each of the quantum readyservice 603 and the classical service 604 may be directed to the arbiter602 (or another arbiter). Next, an indication of a solution to thecomputational task may be provided to the client or the user, such asbeing directed to a user interface of an electronic device of the clientor the user (e.g., being directed over a network, such as over thecloud). The solution may comprise individual solutions to thesub-problems. The indication may include the solution or the individualsolutions.

The technology disclosed herein may comprise a series of sub-processesthat may involve intelligently decomposing a hard (e.g., complex)computational task into simpler (e.g., less complex) sub-problems. Thesystem may further intelligently decide how to distribute the decomposedtasks between a plurality of classical computation resources andquantum-ready computation services.

Referring again to FIG. 6, the quantum-enabled API gateway 601 maycomprise a user-facing service responsible for providing one or more ofthe following: Authentication, Monitoring (e.g., logging), and Bandwidththrottling. The user-facing service may comprise a programmatic accessto a client computer. The authentication may check the identity of auser and determine if the access to the quantum-enabled resources shouldbe granted.

Referring again to FIG. 6, the arbiter 602 may solve quantum problemsand classical problems together. In some applications, the arbiter 602may decompose a given problem using an intelligent algorithm. Thearbiter 602 may comprise one or more intelligent algorithms operating ina centralized or distributed classical processing environment. Thearbiter may provide a quantum-enabled software service by operating oneor more of the following: (1) Breaking down (e.g., decomposing) a givenproblem into sub-problems; (2) Identifying the sub-problems that can besolved using a quantum-ready service 603; (3) Distributing tasks betweenthe classical and quantum-ready services 603 and 604, respectively,accordingly; (4) Collecting solutions of the sub-problems from theclassical and quantum-ready services 603 and 604, respectively; (5)Reducing the original computational tasks using the collected solutionsto sub-problems; (6) If the original problem is completely solved, thesystem may provide an indication of the solution and terminate;otherwise, the system may repeat operation (1) for the remaining portionof the reduced problem. The operations of quantum-ready service 603 maybe based on the technologies described elsewhere herein. On the otherhand, classical service 604 may comprise any cloud-based softwareservice configured to address processing of expensive computationaltasks by obtaining an indication of such tasks from a client; applyingrequired processes to transform the indication of such tasks to a properform; and submitting the indication of such tasks to one or moreclassical digital computing devices, such as computers, clusters ofcomputers, supercomputers, etc.

In various implementations, a computing system may include parallel ordistributed computing. The quantum-ready service 603 and classicalservice 604 may operate in parallel. Further, parallel computing may beimplemented in the quantum-ready service 603. For example, referring toFIG. 5, a quantum computer may solve multiple computational problems inparallel in the worker farm 151; a single problem or sub-problem may besolved in parallel in the worker farm 151. Similarly, a classicalcomputer may solve multiple computational problems in parallel; a singleproblem or sub-problem may be further solved in a parallel ordistributed manner.

Intelligent algorithms for decomposition and distribution may be dynamicand problem dependent. One or more such intelligent algorithms may beused. Referring to FIG. 7, intelligent algorithms may model a feasiblesolution space as a search tree 701. Each node of the tree may be usedto decompose an original computing problem into correspondingsub-problems including disjoint or overlapping sets of variables. Intree 701, different nodes represent a sub-problem solver; for instance,nodes 0, 2, 5, 6, and 11-14 may be capable of solving classical tasks,while nodes 1, 3-4 and 7-10 may be capable of solving quantum tasks.Further, a capability of solving classical and quantum tasks may varyfrom node to node; for example, nodes 3 and 8 may be able to solve fewclassical tasks and many quantum tasks, while node 7 may be able tosolve many classical tasks and few quantum tasks. The intelligentalgorithms may compute certain characteristics of the potentialsub-problems at a certain node in the search tree. Examples ofcharacteristics may include, but are not limited to, adequacy inclassical solvers, adequacy in quantum solvers, complexity (e.g., timeand processor cycles) of computing tasks, current computing capacity inquantum and classical sources, and an estimated time of computedsolutions. The characteristics may be deterministic or probabilisticallymodeled. The intelligent algorithms may have access to information aboutthe size restrictions, capacity, and best-case performance modes of eachof the available quantum and classical computing resources. Theintelligent algorithms may use information available about the quantumand classical computing resources as well characteristics of potentialsub-problems. The intelligent algorithms can determine whether it isadvantageous to decompose the problem at a certain node of the searchtree. If a decomposition takes place, the resulting sub-problems may beadded to the pool of sub-problems together with their correspondingnodes in the search tree. If a decomposition is not advantageous, theintelligent algorithms may continue traversing the search treeconsidering all the possible nodes, until a certain decomposition isadvantageous. Based on partial results of sub-problems received from thequantum or classical computing resources, the intelligent algorithms maybe able to reduce the search tree by pruning certain nodes which may notcontribute to a better solution.

Although the present disclosure has made reference to quantum computers,methods and systems of the present disclosure may be employed for usewith other types of computers, which may be non-classical computers.Such non-classical computers may comprise quantum computers, hybridquantum computers, quantum-type computers, or other computers that arenot classical computers. Examples of non-classical computers mayinclude, but are not limited to, Hitachi Ising solvers, coherent Isingmachines based on optical parameters, and other solvers which utilizedifferent physical phenomena to obtain more efficiency in solvingparticular classes of problems.

Transactions

In various embodiments, the systems, methods, platforms, and mediadescribed herein may comprise a transactional unit for receiving an itemof value in exchange for at least executing the one or more instructionsto generate the one or more solutions. The item of value may comprisemoney or credit. The item of value may be received from a user of thedigital computer. The transactional unit may determine a cost forexecuting the one or more instructions to generate the one or moresolutions. The transactional unit may determine the cost after or priorto executing the one or more instructions, and wherein the one or moreinstructions may be executed upon receiving authorization to execute theone or more instructions. The authorization may be received from a userof the digital computer. The item of value may be equal to the cost.

Methods and Systems for Generating Programming Problems

In another aspect, the present disclosure provides methods and systemsfor generating a programming problem. The programming problem may besolved using a non-classical computer of the present disclosure, suchas, for example, a quantum computer.

The method disclosed herein can be applied to any quantum system ofsuperconducting qubits, comprising local field biases on the qubits, anda plurality of couplings of the qubits, and control systems for applyingand tuning local field biases and coupling strengths. Systems of quantumdevices as such are disclosed for instance in US20120326720 andUS20060225165.

Disclosed invention comprises a method for finding an integer encodingthat uses the minimum number of binary variables in representation of aninteger variable, while respecting an upper bound on the values ofcoefficients appearing in the encoding. Such an encoding is referred toas a “bounded-coefficient encoding”. It also comprises a method forproviding a system of constraints on the binary variables to preventdegeneracy of the bounded-coefficient encoding. Such a system ofconstraints involving the binary variables is referred to as “a systemof non-degeneracy constraints”.

Disclosed invention further comprises of employing bounded-coefficientencoding to represent a polynomial on a bounded integer domain as theHamiltonian of a system of superconducting qubits.

An advantage of the method disclosed herein is that it enables anefficient method for finding the solution of a mixed integerpolynomially constrained polynomial programming problem by finding thesolution of an equivalent binary polynomially constrained polynomialprogramming. In one embodiment, the equivalent binary polynomiallyconstrained polynomial programming problem might be solved by a systemof superconducting qubits as disclosed in U.S. Ser. No. 15/051,271, U.S.Ser. No. 15/014,576, CA2921711 and CA2881033. Methods and systems forgenerating programming problems of the present disclosure may be asdescribed in 2017/0344898 (“Methods and systems for setting a system ofsuper conducting qubits having a Hamiltonian representative of apolynomial on a bounded integer domain”, which is entirely incorporatedherein by reference.

Described herein is a method for setting a system of superconductingqubits having a Hamiltonian representative of a polynomial on a boundedinteger domain via bounded-coefficient encoding, the method comprising:using one or more computer processors to obtaining (i) the polynomial onthe bounded integer domain and (ii) integer encoding parameters;computing the bounded-coefficient encoding using the integer encodingparameters; recasting each integer variable as a linear function ofbinary variables using the bounded-coefficient encoding, and providingadditional constraints on the attained binary variables to avoiddegeneracy in the encoding, if required by a user; substituting eachinteger variable with an equivalent binary representation, and computingthe coefficients of the equivalent binary representation of thepolynomial on the bounded integer domain; performing a degree reductionon the obtained equivalent binary representation of the polynomial onthe bounded integer domain to provide an equivalent polynomial of degreeat most two in binary variables; and setting local field biases andcoupling strengths on the system of superconducting qubits using thecoefficients of the derived polynomial of degree at most two in severalbinary variables.

Also described herein, in certain embodiments, is a system comprising: asub-system of superconducting qubits; a computer operatively coupled tothe sub-system of superconducting qubits, wherein the computer comprisesat least one computer processor, an operating system configured toperform executable instructions, and a memory; and a computer programincluding instructions executable by the at least one computer processorto generate an application for setting the sub-system of superconductingqubits having a Hamiltonian representative of a polynomial on a boundedinteger domain via bounded-coefficient encoding, the applicationcomprising: a first module programmed or otherwise configured to obtainthe polynomial on the bounded integer domain; a second module programmedor otherwise configured to obtain integer encoding parameters; a thirdmodule programmed or otherwise configured to compute abounded-coefficient encoding using the integer encoding parameters; afourth module programmed or otherwise configured to recast each integervariable as a linear function of binary variables using thebounded-coefficient encoding and provide additional constraints on theattained binary variables to avoid degeneracy in the encoding, ifrequired by a user; a fifth module programmed or otherwise configured tosubstitute each integer variable with an equivalent binaryrepresentation, and compute the coefficients of the equivalent binaryrepresentation of the polynomial on the bounded integer domain; asoftware module programmed or otherwise configured to reduce apolynomial in several binary variables to a polynomial of degree at mosttwo in several binary variables; a software module programmed orotherwise configured to provide an assignment of binary variables of theobtained equivalent binary polynomial of degree at most two to qubits;and a software module programmed or otherwise configured to set localfield biases and coupling strengths on the system of superconductingqubits using the coefficients of the derived polynomial of degree atmost two in several binary variables.

Also described herein, in certain embodiments, is a non-transitorycomputer-readable medium comprising machine-executable code that, uponexecution by one or more computer processors, implements a method forsetting a system of superconducting qubits having a Hamiltonianrepresentative of a polynomial on a bounded integer domain viabounded-coefficient encoding, the method comprising: using one or morecomputer processors to obtain (i) the polynomial on the bounded integerdomain and (ii) integer encoding parameters; computing thebounded-coefficient encoding using the integer encoding parameters;recasting each integer variable as a linear function of binary variablesusing the bounded-coefficient encoding, and providing additionalconstraints on the attained binary variables to avoid degeneracy in theencoding, if required by a user; substituting each integer variable withan equivalent binary representation, and computing the coefficients ofthe equivalent binary representation of the polynomial on the boundedinteger domain; performing a degree reduction on the obtained equivalentbinary representation of the polynomial on the bounded integer domain toprovide an equivalent polynomial of degree at most two in binaryvariables; and setting local field biases and coupling strengths on thesystem of superconducting qubits using the coefficients of the derivedpolynomial of degree at most two in several binary variables.

Various Definitions

Unless otherwise defined, all technical terms used herein have the samemeaning as commonly understood by one of ordinary skill in the art towhich this invention belongs. As used in this specification and theappended claims, the singular forms “a,” “an,” and “the” include pluralreferences unless the context clearly dictates otherwise. Any referenceto “or” herein is intended to encompass “and/or” unless otherwisestated.

The term “integer variable” and like terms refer to a data structure forstoring integers in a digital system, between two integers l and u wherel≤u. The integer l is called the “lower bound” and the integer u iscalled the “upper bound” of the integer variable x.

It is appreciated by the skilled addressee that every integer variable xwith lower and upper bounds l and u respectively, can be transformed toa bounded integer variable x with lower and upper bounds 0 and u−lrespectively.

Accordingly, herein the term “bounded integer variable” refers to aninteger variable which may represent integer values with lower boundequal to 0. One may denote a bounded integer variable x with upper boundu by x∈{0, 1, . . . , u}.

The term “binary variable” and like terms refer to a data structure forstoring integers 0 and 1 in a digital system. It is appreciated that insome embodiments computer bits are used to store such binary variables.

The term “integer encoding” of a bounded integer variable x refers to atuple (c₁, . . . , c_(d)) of integers such that the identity x=Σ_(i=1)^(d) c_(i) y _(i) is satisfied for every possible value x of x using achoice of binary numbers y ₁, . . . , y_(d) for binary variables y₁, . .. , y_(d).

The term “bounded-coefficient encoding” with bound M, refers to aninteger encoding (c₁, . . . , c_(d)) of a bounded integer variable xsuch that c_(i)≤M for all i=1, . . . , d, and uses the least number ofbinary variables y₁, . . . , y_(d) amongst all encodings of x satisfyingthese inequalities.

The term “a system of non-degeneracy constraints” refers to a system ofconstraints that makes the equation x=Σ_(i=1) ^(d) c_(i) y _(i) have aunique binary solution (y ₁, . . . , y _(d)) for every choice of value xfor variable x.

The term “polynomial on a bounded integer domain” and like terms mean afunction of the form

${{f(x)} = {\sum\limits_{t = 1}^{T}\;{Q_{t}{\prod\limits_{i = 1}^{n}\; x_{i}^{p_{i}^{t}}}}}},$in several integer variables x_(i)∈{0, 1, 2, . . . , κ_(i)} for i=1, . .. , n, where p_(i) ^(t)≥0 is an integer denoting the power of variablex_(i) in t-th term and κ_(i) is the upper bound of x_(i).

The term “polynomial of degree at most two on bounded integer domain”and like terms mean a function of the form

${f(x)} = {{\sum\limits_{i,{j = 1}}^{n}\;{Q_{ij}x_{i}x_{j}}} + {\sum\limits_{i = 1}^{n}\;{q_{i}x_{i}}}}$in several integer variables x_(i)∈{0, 1, 2, . . . , κ_(i)} for i=1, . .. , n, where κ_(i) is the upper bound of x_(i).

It is appreciated that a polynomial of degree at most two on binarydomain, can be represented by a vector of linear coefficients (q₁, . . ., q_(n)) and an n×n symmetric matrix Q=(Q_(ij)) with zero diagonal.

The term “mixed-integer polynomially constrained polynomial programming”problem and like terms mean finding the minimum of a polynomial y=ƒ(x)in several variables x=(x₁, . . . , x_(n)), that a nonempty subset ofthem indexed by S⊆{1, . . . , n} are bounded integer variables and therest are binary variables, subject to a (possibly empty) family ofequality constraints determined by a (possibly empty) family of eequations g_(j)(x)=0 for j=1, . . . , e and a (possibly empty) family ofinequality constraints determined by a (possibly empty) family of linequalities h_(j)(x)≤0 for j=1, . . . , l. Here, all functions ƒ(x),g_(i)(x) for i=1, . . . , e and h_(j)(x) for j=1, . . . , l arepolynomials. It is appreciated that a mixed integer polynomiallyconstrained polynomial programming problem can be represented as:min ƒ(x)subject to g _(i)(x)=0 ∀i∈{1, . . . ,e},h _(j)(x)≤0 ∀j∈{1, . . . ,l},x _(s)∈{0, . . . ,κ_(s) } ∀s∈S⊆{1, . . . ,n},x _(s)∈{0,1} ∀s∉S.The above mixed integer polynomially constrained polynomial programmingproblem will be denoted by (P_(I)) and the optimal value of it will bedenoted by ν(P_(I)). An optimal solution, denoted by x*, is a vector atwhich the objective function attains the value ν(P_(I)) and allconstraints are satisfied.

The term “polynomial of degree at most two on binary domain” and liketerms mean a function of form ƒ(x)=Σ_(i,j=1)^(n)Q_(ij)x_(i)x_(j)+Σ_(i=1) ^(n)q_(i)x_(i) defined on several binaryvariables x_(i)∈{0,1} for i=1, . . . , n.

It is appreciated that a polynomial of degree at most two on binarydomain, can be represented by a vector of linear coefficients (q₁, . . ., q_(n)) and a n×n symmetric matrix Q=(Q_(ij)) with zero diagonal.

The term “binary polynomially constrained polynomial programming”problem and like terms mean a mixed-integer polynomially constrainedpolynomial programming P_(I) such that S=Ø:min ƒ(x)subject to g _(i)(x)=0 ∀i∈{1 . . . ,e}h _(j)(x)≤0 ∀j∈{1, . . . l}x _(k)∈{0,1} ∀k∈{1, . . . n}.The above binary polynomially constrained polynomial programming problemis denoted by P_(B) and its optimal value is denoted by ν(P_(B)).

Two mathematical programming problems are called “equivalent” if havingthe optimal solution of each one of them, the optimal solution of theother one can be computed in polynomial time of the size of formeroptimal solution.

The term “qubit” and like terms generally refer to any physicalimplementation of a quantum mechanical system represented on a Hilbertspace and realizing at least two distinct and distinguishableeigenstates representative of the two states of a quantum bit. A quantumbit is the analogue of the digital bits, where the ambient storingdevice may store two states |0

and |1

of a two-state quantum information, but also in superpositionsα|0

+β|1

of the two states. In various embodiments, such systems may have morethan two eigenstates in which case the additional eigenstates are usedto represent the two logical states by degenerate measurements. Variousembodiments of implementations of qubits have been proposed; e.g. solidstate nuclear spins, measured and controlled electronically or withnuclear magnetic resonance, trapped ions, atoms in optical cavities(cavity quantum-electrodynamics), liquid state nuclear spins, electroniccharge or spin degrees of freedom in quantum dots, superconductingquantum circuits based on Josephson junctions (Barone and Paternò, 1982,“Physics and Applications of the Josephson Effect,” John Wiley and Sons,New York; Martinis et al., 2002, Physical Review Letters 89, 117901) andelectrons on Helium.

The term “local field”, refers to a source of bias inductively coupledto a qubit. In one embodiment a bias source is an electromagnetic deviceused to thread a magnetic flux through the qubit to provide control ofthe state of the qubit (U.S. Patent Publication No. 20060225165].

The term “local field bias” and like terms refer to a linear bias on theenergies of the two states |0

and |1

of the qubit. In one embodiment the local field bias is enforced bychanging the strength of a local field in proximity of the qubit (U.S.Patent Publication No. US2006/0225165).

The term “coupling” of two qubits H₁ and H₂ is a device in proximity ofboth qubits threading a magnetic flux to both qubits. In one embodiment,a coupling may consist of a superconducting circuit interrupted by acompound Josephson junction. A magnetic flux may thread the compoundJosephson junction and consequently thread a magnetic flux on bothqubits (U.S. Patent Publication No. 2006/0225165).

The term “coupling strength” between qubits H₁ and H₂ refer to aquadratic bias on the energies of the quantum system comprising bothqubits. In one embodiment the coupling strength is enforced by tuningthe coupling device in proximity of both qubits.

The term “quantum device control system”, refers to a system comprisinga digital processing unit capable of initiating and tuning the localfield biases and couplings strengths of a quantum system.

The term “system of superconducting qubits” and like, refers to aquantum mechanical system comprising a plurality of qubits and pluralityof couplings between a plurality of pairs of the plurality of qubits,and further comprising a quantum device control system.

It is appreciated that a system of superconducting qubits may bemanufacture in various embodiments. In one embodiment a system ofsuperconducting qubits is a “quantum annealer”.

The term “quantum annealer” and like terms mean a system ofsuperconducting qubits that carries optimization of a configuration ofspins in an Ising spin model using quantum annealing as described, forexample, in Farhi, E. et al., “Quantum Adiabatic Evolution Algorithmsversus Simulated Annealing” arXiv.org: quant ph/0201031 (2002), pp.1-16. An embodiment of such an analog processor is disclosed by McGeoch,Catherine C. and Cong Wang, (2013), “Experimental Evaluation of anAdiabatic Quantum System for Combinatorial Optimization” ComputingFrontiers,” May 14-16, 2013(http://www.cs.amherst.edu/ccm/cf14-mcgeoch.pdf) and also disclosed inthe U.S. Patent Application No. US 2006/0225165.

Steps and Architecture for Setting a System of Superconducting Qubits

In some embodiments, the methods, systems, and media described hereininclude a series of steps for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding. In some embodiments, the methoddisclosed herein can be used in conjunction with any method on anysolver for solving a binary polynomially constrained polynomialprogramming problem to solve a mixed-integer polynomially constrainedpolynomial programming problem.

Referring to FIG. 8, in a particular embodiment, a flowchart of allsteps is presented as for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain. Specifically, processing step 802 is shown to be obtaining aplurality of integer variables on a bounded integer domain and anindication for a polynomial in those variables. Processing step 804 isdisclosed as for obtaining integer encoding parameters. Processing step806 is used to be computing a bounded-coefficient encoding of theinteger variable(s) and the system of non-degeneracy constraints.Processing step 808 is displayed to be obtaining a polynomial in severalbinary variables equivalent to the provided polynomial on a boundedinteger domain. Processing step 810 is shown to be performing a degreereduction on the obtained polynomial in several binary variables toprovide a polynomial of degree at most two in several binary variables.Processing step 812 is shown to be providing an assignment of binaryvariables of the equivalent polynomial of degree at most two to qubits.Processing step 812 is used to be setting local field biases andcouplings strengths.

Referring to FIG. 9, in a particular embodiment, a diagram of a systemfor setting a system of superconducting qubits having a Hamiltonianrepresentative of a polynomial on a bounded integer domain isdemonstrated to be comprising of a digital computer interacting with asystem of superconducting qubits.

Specially, there is shown an embodiment of a system 900 in which anembodiment of the method for setting a system of superconducting qubitsin such a way that its Hamiltonian is representative of a polynomial ona bounded integer domain may be implemented. The system 900 comprises adigital computer 902 and a system 904 of superconducting qubits. Thedigital computer 902 receives a polynomial on a bounded integer domainand the encoding parameters and provides the bounded-coefficientencoding, a system of non-degeneracy constraints, and the values oflocal fields and couplers for the system of superconducting qubits.

It will be appreciated that the polynomial on a bounded integer domainmay be provided according to various embodiments. In one embodiment, thepolynomial on a bounded integer domain is provided by a user interactingwith the digital computer 902. Alternatively, the polynomial on abounded integer domain is provided by another computer, not shown,operatively connected to the digital computer 902. Alternatively, thepolynomial on a bounded integer domain is provided by an independentsoftware package. Alternatively, the polynomial on a bounded integerdomain is provided by an intelligent agent.

It will be appreciated that the integer encoding parameters may beprovided according to various embodiments. In one embodiment, theinteger encoding parameters are provided by a user interacting with thedigital computer 202. Alternatively, the integer encoding parameters areprovided by another computer, not shown, operatively connected to thedigital computer 902. Alternatively, the integer encoding parameters areprovided by an independent software package. Alternatively, the integerencoding parameters are provided by an intelligent agent.

In some embodiments, the skilled addressee appreciates that the digitalcomputer 202 may be any type. In one embodiment, the digital computer202 is selected from a group consisting of desktop computers, laptopcomputers, tablet PCs, servers, smartphones, etc.

Referring to FIG. 10, in a particular embodiment, a diagram of a systemfor setting a system of superconducting qubits having a Hamiltonianrepresentative of a polynomial on a bounded integer domain isdemonstrated to be comprising of a digital computer used for computingthe local fields and couplers.

Further referring to FIG. 10, there is shown an embodiment of a digitalcomputer 902 interacting with a system 904 of superconducting qubits. Itwill be appreciated that the digital computer 902 may also be broadlyreferred to as a processor. In this embodiment, the digital computer 902comprises a central processing unit (CPU) 1002, also referred to as amicroprocessor, a display device 1004, input devices 1006, communicationports 1008, a data bus 1010, a memory unit 1012 and a network interfacecard (NIC) 1022.

The CPU 1002 is used for processing computer instructions. It'sappreciated that various embodiments of the CPU 1002 may be provided. Inone embodiment, the central processing unit 1002 is a CPU Core i7-3820running at 3.6 GHz and manufactured by Intel™.

The display device 1004 is used for displaying data to a user. Theskilled addressee will appreciate that various types of display device1004 may be used. In one embodiment, the display device 1004 is astandard liquid-crystal display (LCD) monitor.

The communication ports 1008 are used for sharing data with the digitalcomputer 902. The communication ports 1008 may comprise, for instance, auniversal serial bus (USB) port for connecting a keyboard and a mouse tothe digital computer 902. The communication ports 1008 may furthercomprise a data network communication port such as an IEEE 802.3 portfor enabling a connection of the digital computer 902 with anothercomputer via a data network. The skilled addressee will appreciate thatvarious alternative embodiments of the communication ports 1008 may beprovided. In one embodiment, the communication ports 308 comprise anEthernet port and a mouse port (e.g., Logitech™).

The memory unit 1012 is used for storing computer-executableinstructions. It will be appreciated that the memory unit 1012comprises, in one embodiment, an operating system module 1014. It willbe appreciated by the skilled addressee that the operating system module1014 may be of various types. In an embodiment, the operating systemmodule 1014 is OS X Yosemite manufactured by Apple™.

The memory unit 1012 further comprises an application for providing apolynomial on a bounded integer domain, and integer encoding parameters1016. The memory unit 1012 further comprises an application for reducingthe degree of a polynomial in several binary variables to at most two1018. It is appreciated that the application for reducing the degree ofa polynomial in several binary variables can be of various kinds. Oneembodiment of an application for reducing degree of a polynomial inseveral binary variables to at most two is disclosed by H. Ishikawa,“Transformation of General Binary MRF Minimization to the First-OrderCase,” in IEEE Transactions on Pattern Analysis and MachineIntelligence, vol. 33, no. 6, pp. 1234-1249, June 2011 and by MartinAnthony, Endre Boros, Yves Crama, and Aritanan Gruber. 2016.Quadratization of symmetric pseudo-Boolean functions. Discrete Appl.Math. 203, C (April 2016), 1-12.(DOI=http://dx.doi.org/10.1016/j.dam.2016.01.001). The memory unit 1012further comprises an application for minor embedding of a source graphto a target graph 1020. It is appreciated that the application for minorembedding can be of various kinds. One embodiment of an application forminor embedding of a source graph to a target graph is disclosed in U.S.Pat. No. 8,244,662. The memory unit 1012 further comprises anapplication for computing the local field biases and couplingsstrengths.

Each of the central processing unit 1002, the display device 1004, theinput devices 1006, the communication ports 1008 and the memory unit1012 is interconnected via the data bus 1010.

The system 902 further comprises a network interface card (NIC) 1022.The application 1020 sends the appropriate signals along the data bus1010 into NIC 1022. NIC 1022, in turn, sends such information to quantumdevice control system 1024.

The system 904 of superconducting qubits comprises a plurality ofsuperconducting quantum bits and a plurality of coupling devices.Further description of such a system is disclosed in U.S. PatentApplication No. 2006/0225165.

The system 904 of superconducting qubits, further comprises a quantumdevice control system 1024. The control system 1024 itself comprisescoupling controller for each coupling in the plurality 328 of couplingsof the device 204 capable of tuning the coupling strengths of acorresponding coupling, and local field bias controller for each qubitin the plurality 1026 of qubits of the device 904 capable of setting alocal field bias on each qubit.

Obtaining a Plurality of Integer Variables on a Bounded Integer Domain

In some embodiments, the methods, systems, and media described hereininclude a series of steps for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding. In some embodiments, aprocessing step is shown to be obtaining a plurality of integervariables on a bounded integer domain and an indication for a polynomialin those variables.

Referring to FIG. 8 and according to processing step 802, a polynomialon a bounded integer domain is obtained. Referring to FIG. 11, in aparticular embodiment, there is shown of a detailed processing step forproviding a polynomial on a bounded integer domain.

According to processing step 1102, the coefficient of each term of apolynomial and the degree of each variable in the corresponding term areprovided. It is appreciated that providing the coefficient and degree ofeach variable in each term can be performed in various embodiments. Inone embodiments a list of form [Q_(t), p₁ ^(t), p₂ ^(t), . . . p_(n)^(t)] is provided for each term of the polynomial in which Q_(t) is thecoefficient of the t-th term and p_(i) ^(t) is the power of i-thvariable in the t-th term.

In another embodiment, and in the particular case that the providedpolynomial is of degree at most two a list (q₁, . . . , q_(n)) and a n×nsymmetric matrix Q=(Q_(ij)) is provided. It is appreciated that a singlebounded integer variable is an embodiment of a polynomial of degree atmost two in which n=1, q₁=1 and Q=(Q₁₁)=(0).

It the same embodiment, it is also appreciated that if Q_(ij)=0 for alli,j=1, . . . , n, the provided polynomial is a linear function.

It will be appreciated that the providing of a polynomial may beperformed according to various embodiments.

As mentioned above and in one embodiment, the coefficients of apolynomial are provided by a user interacting with the digital computer902. Alternatively, the coefficients of a polynomial are provided byanother computer operatively connected to the digital computer 902.Alternatively, the coefficients of a polynomial are provided by anindependent software package. Alternatively, an intelligent agentprovides the coefficients of a polynomial.

According to processing step 1104, an upper bound on each boundedinteger variable is provided. It will be appreciated that the providingof upper bounds on the bounded integer variables may be performedaccording to various embodiments.

As mentioned above and in one embodiment, the upper bounds on theinteger variables are provided by a user interacting with the digitalcomputer 902. Alternatively, the upper bounds on the integer variablesare provided by another computer operatively connected to the digitalcomputer 902. Alternatively, the upper bounds on the integer variablesare provided by an independent software package or a computer readableand executable subroutine. Alternatively, an intelligent agent providesthe upper bounds on the integer variables.

Obtaining Integer Encoding Parameters

In some embodiments, the methods, systems, and media described hereininclude a series of steps for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding. In some embodiments, aprocessing step is shown to be obtaining integer encoding parameters.Referring to FIG. 8 and processing step 804, the integer encodingparameters are obtained.

It is appreciated that the integer encoding parameters comprises ofeither obtaining an upper bound on the coefficients c_(i)'s of thebounded-coefficient encoding directly; or obtaining the error tolerances∈_(l) and ∈_(c) of the local field biases and couplings strengths,respectively. If the upper bound on the coefficients c_(i)'s is notprovided directly, it is computed by the digital computer 902 asdescribed in the processing step 1204.

Referring to FIG. 12 and according to processing step 1202, an upperbound on the coefficients of the bounded-coefficient encoding may beprovided. It will be appreciated that the providing of the upper boundon the coefficients of the bounded-coefficient encoding may be performedaccording to various embodiments. In one embodiment, the upper bound onthe coefficients of the bounded-coefficient encoding is provideddirectly by a user, a computer, a software package or an intelligentagent.

Still referring to processing step 1202, if the upper bound on thecoefficients of the bounded-coefficient encoding is not directlyprovided, the error tolerances of the local field biases and thecouplings strengths of the system of superconducting qubits areprovided. It will be appreciated that the providing of the errortolerances of the local field biases and the couplings strengths of thesystem of superconducting qubits may be performed according to variousembodiments. In one embodiment, the error tolerances of the local fieldbiases and the couplings strengths of the system of superconductingqubits are provided directly by user, a computer, a software package oran intelligent agent.

According to processing steps 1204, the upper bound on the coefficientsof the bounded-coefficient encoding is obtained based on the errortolerances ∈_(l) and ∈_(c) respectively of the local field biases andcouplings strengths of the system of superconducting qubits.

Still referring to processing step 1204 the upper bound of the values ofthe coefficients of the integer encoding is obtained. The description ofthe system used for computing the upper bound of the coefficients of thebounded-coefficient encoding when ∈_(l) and ∈_(c) are provided, is nowpresented in detail.

If the provided polynomial is only a single bounded integer variable x,then the upper bound on the coefficients of the bounded-coefficientencoding of x denoted by μ^(x) is computed and stored as

$\mu^{x} = {\left\lfloor \frac{1}{\epsilon_{\ell}} \right\rfloor.}$

If the provided polynomial is of degree one, i.e. ƒ(x)=Σ_(i=1)^(n)q_(i)x_(i), then the upper bound of the coefficients of thebounded-coefficient encoding for variable x_(i) is computed and storedas

$\mu^{x_{i}} = {\left\lfloor \frac{\min\limits_{j}\left\{ {q_{j}} \right\}}{{q_{i}}\epsilon_{\ell}} \right\rfloor.}$

It is appreciated that if μ^(x) ^(i) for i=1, . . . , n are required tobe of equal value, the upper bound of the coefficients of thebounded-coefficient encoding is stored as

$\mu = {\left\lfloor \frac{\min\limits_{j}\left\{ {q_{j}} \right\}}{\max\limits_{j}{\left\{ {q_{i}} \right\}\epsilon_{\ell}}} \right\rfloor.}$It is appreciated that this value of μ coincides with min_(j){μ^(x) ^(j)}.

If the provided polynomial is of degree at least two, i.e., ƒ(x)=Σ_(t=1)^(T)Q_(t)Π_(i=1) ^(n)x_(i) ^(p) ^(i) ^(t) , and there exists a t suchthat Σ_(i=1) ^(n)p_(i) ^(t)≥2, the upper bounds on the coefficients ofthe bounded-coefficient encodings for variables x_(i) for i=1, . . . , nmust be such that the coefficient of the equivalent polynomial of degreeat most two in several variables derived after the substitution ofbinary representation of x_(i)'s and performing the degree reduction,i.e.,

${f(x)} = {{\sum\limits_{i = 1}^{\overset{\_}{n}}{q_{i}^{B}y_{i}}} + {\sum\limits_{\underset{i \neq j}{i,{j = 1}}}^{\overset{\_}{n}}{Q_{ij}^{B}y_{i}y_{j}}}}$satisfy the following inequalities:

${\frac{\min\limits_{i}{q_{i}^{B}}}{\max\limits_{i}{q_{i}^{B}}} \geq \epsilon_{\ell}},{and}$$\frac{\min\limits_{i}{Q_{ij}^{B}}}{\max\limits_{i}{Q_{ij}^{B}}} \geq {\epsilon_{c}.}$

It is appreciated that finding the upper bounds on the coefficients ofthe bounded-coefficient encoding such that the above inequalities aresatisfied can be done in various embodiments. In one embodiment, avariant of a bisection search may be employed to find the upper boundson the coefficients of the bounded-coefficient encoding such that theabove inequalities are satisfied. In another embodiment a suitableheuristic search utilizing the coefficients and degree of the polynomialmay be employed to find the upper bounds on the coefficients of thebounded-coefficient encoding such that the above inequalities aresatisfied.

In a particular case that ƒ(x)=Σ_(i=1) ^(n)q_(i)x_(i)+Σ_(i,j=1)^(n)Q_(ij)x_(i)x_(j), and Q_(ii) and q_(i) are of the same sign, theabove set of inequalities are reduced to

${{{{{Q_{ii}}\left( \mu^{x_{i}} \right)^{2}} + {{q_{i}}\mu^{x_{i}}}} \leq {\frac{m_{\ell}}{\epsilon_{\ell}}\mspace{14mu}{for}\mspace{14mu} i}} = 1},\ldots\mspace{14mu},n,{{\mu^{x_{i}} \leq {\sqrt{\frac{m_{c}}{{Q_{ii}}\epsilon_{c}}}\mspace{14mu}{for}\mspace{14mu} i}} = 1},\ldots\mspace{14mu},{{n\text{:}\mspace{14mu} Q_{ii}} \neq 0},{{\mu^{x_{i}}\mu^{x_{j}}} \leq {\frac{m_{c}}{{Q_{ij}}\epsilon_{c}}\mspace{14mu}{for}\mspace{14mu} i}},{j = 1},\ldots\mspace{14mu},{{n\text{:}\mspace{14mu} Q_{ij}} \neq 0},$for m_(l)=min_(i){|Q_(ii)+q_(i)|} and m_(c)=min_(i,j){|Q_(ii)|,|Q_(ij)|}. Various methods may be employed to find μ^(x) ^(i) for i=1, .. . , n that satisfy the above set of inequalities. In one embodimentthe following mathematical programming model may be solved withappropriate solver on the digital computer 202 to find μ^(x) ^(i) fori=1, . . . , n.

${\min{\sum\limits_{i = 1}^{n}\frac{\kappa^{x_{i}}}{\mu^{x_{i}}}}},{{{{{subject}\mspace{14mu}{to}\mspace{14mu}{Q_{ii}\left( \mu^{x_{i}} \right)}^{2}} + {q_{i}\mu^{x_{i}}}} \leq {\frac{m_{\ell}}{\epsilon_{\ell}}\mspace{14mu}{for}\mspace{14mu} i}} = 1},\ldots\mspace{14mu},n,{{\mu^{x_{i}} \leq {\sqrt{\frac{m_{c}}{{Q_{cc}}\epsilon_{c}}}\mspace{14mu}{for}\mspace{14mu} i}} = 1},{{\ldots\mspace{14mu} n\text{:}\mspace{14mu} Q_{ii}} \neq 0},{{\mu^{x_{i}}\mu^{x_{j}}} \leq {\frac{m_{c}}{{Q_{ij}}\epsilon_{c}}\mspace{14mu}{for}\mspace{14mu} i}},{j = 1},{{\ldots\mspace{14mu} n\text{:}\mspace{14mu} Q_{ij}} \neq 0.}$In another embodiment a heuristic search algorithm may be employed forfinding μ^(x) ^(i) for i=1, 2, . . . , n that satisfy the aboveinequalities.Computing a Bounded-Coefficient Encoding of the Integer Variable(s) andthe System of Non-Degeneracy Constraints

In some embodiments, the methods, systems, and media described hereininclude a series of steps for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding. In some embodiments, aprocessing step is shown to be computing a bounded-coefficient encodingof the integer variable(s) and the system of non-degeneracy constraints.Referring to FIG. 8 and processing step 806, the bounded-coefficientencoding and the system of non-degeneracy constraints are obtained.

Referring to FIG. 13, in a particular embodiment, it explains how thebounded-coefficient encoding is derived. Herein, it denotes the upperbound on the integer variable x with κ^(x), and the upper bound on thecoefficients used in the integer encoding with μ^(x). According toprocessing step 1302, it derives the binary encoding of μ^(x). It setsl_(μ) _(x) =└ log₂ μ^(x)+1 ┘. Then the binary encoding of μ is set to beS _(μ) _(x) =(2^(i-1):for i=1, . . . ,l _(μ) _(x) ).It is appreciated that if κ^(x)<2^(l) ^(μ) ^(x) then the binary encodingof κ^(x) does not have any coefficients larger than μ^(x) and theprocessing step 1302 derives

$c_{i}^{x} = \left\{ {\begin{matrix}2^{i - 1} & {{{{for}\mspace{14mu} i} = 1},\ldots\mspace{14mu},\left\lfloor {\log_{2}\kappa^{x}} \right\rfloor} \\{\kappa^{x} - {\sum\limits_{i = 1}^{\lfloor{{lo}\; g_{2}\kappa^{x}}\rfloor}2^{i - 1}}} & {{{for}\mspace{14mu} i} = {\left\lfloor {\log_{2}\kappa^{x}} \right\rfloor + 1}}\end{matrix},} \right.$and processing step 604 is skipped.

Still referring to FIG. 13 and according to processing step 1304, itcompletes the bounded-coefficient encoding if required (i.e.,κ^(x)≥2^(l) ^(μ) ^(x)) by adding

$\eta_{\mu^{x}} = \left\lfloor \frac{\left( {\kappa^{x} - {\sum\limits_{i = 0}^{\ell_{\mu^{x}}}2^{i - 1}}} \right)}{\mu^{x}} \right\rfloor$coefficients of value μ, and one coefficient of valueτ^(x)=κ^(x)−Σ_(i=0) ^(l) ^(μ) ^(x)2^(i-1)−η_(μ) _(x) μ^(x) if τ isnonzero. Using the derived coefficients, the bounded-coefficientencoding is the integer encoding in which the coefficients are asfollows:

$c_{i}^{x} = \left\{ \begin{matrix}{2^{i - 1},} & {{{{for}\mspace{14mu} i} = 1},\ldots\mspace{14mu},\ell_{\mu^{x}},} \\{\mu^{x},} & {{{{for}\mspace{14mu} i} = {\ell_{\mu^{x}} + 1}},\ldots\mspace{14mu},{\ell_{\mu^{x}} + \eta_{\mu^{x}}},} \\{\tau^{x},} & {{{for}\mspace{14mu} i} = {{{\ell_{\mu}x} + {\eta_{\mu}x} + {1\mspace{14mu}{if}\mspace{14mu}\tau^{x}}} \neq 0.}}\end{matrix} \right.$

It is appreciated that the degree of the bounded-coefficient encoding is

$d^{x} = \left\{ \begin{matrix}{\ell_{\mu^{x}} + \eta_{\mu^{x}}} & {{{{if}\mspace{14mu}\tau^{x}} = 0},} \\{\ell_{\mu^{x}} + \eta_{\mu^{x}} + 1} & {{otherwise}.}\end{matrix} \right.$

It is also appreciated that in the bounded-coefficient encoding thefollowing identity is satisfied

${\sum\limits_{i = 1}^{d^{x}}c_{i\;}^{x}} = {\kappa^{x}.}$

For example, if one needs to encode an integer variable that takesmaximum value of 24 with integer encoding that has maximum coefficientof 6, the bounded-coefficient encoding would bec ₁=1,c ₂=2,c ₃=4,c ₄=6,c ₅=6,c ₅=5.

It is appreciated that bounded-coefficient encoding may be derivedaccording to various embodiment. In one embodiment, it is the output ofa digital computer readable and executable subroutine.

Still referring to FIG. 13 and according to processing step 1306, asystem of non-degeneracy constraints is provided. It is appreciated thatthe system of non-degeneracy constraints may be represented in variousembodiments.

In one embodiment the system of non-degeneracy constraints is thefollowing system of linear inequalities:

${{\sum\limits_{i = 1}^{\ell_{\mu^{x}}}y_{i}^{x}} \geq y_{\ell_{{\mu + 1}\;}}^{x}},{y_{i}^{x} \geq y_{i + 1}^{x}},{{{for}\mspace{14mu} i} = {\ell_{\mu^{x}} + 1}},\ldots\mspace{14mu},{d^{x}.}$

It is appreciated that the providing of the system of non-degeneracyconstraints above may be carried by providing a matrix A of size(d^(x)−l_(μ) _(x) +1)×d^(x) with entries −1, 0, 1. In this embodimentthe system of non-degeneracy constraints is represented by the followingsystem

${A\begin{pmatrix}y_{1}^{x} \\\vdots \\y_{d^{x}}^{x}\end{pmatrix}} \leq {\begin{pmatrix}0 \\\vdots \\0\end{pmatrix}.}$Converting from Integer Domain to Binary Variables

In some embodiments, the methods, systems, and media described hereininclude a series of steps for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding. In some embodiments, aprocessing step is shown to be providing a polynomial in several binaryvariables equivalent to the provide polynomial on a bounded integerdomain. Referring back to FIG. 8 and according to processing step 808,the provided polynomial on a bounded integer domain is converted to anequivalent polynomial in several binary variables.

Referring to FIG. 14 and processing step 1402, each integer variablex_(i) is represented with the following linear function

${x_{i} = {\sum\limits_{k = 1}^{d^{x_{i}}}{c_{k}^{x_{i}}y_{k}^{x_{i}}}}},$of binary variables y_(k) ^(x) ^(i) for k=1, . . . , d^(x) ^(i) .

Still referring to FIG. 14 and according to processing step 1404, thecoefficients of the polynomial on binary variables equivalent to theobtained polynomial on bounded integer domain are computed.

For each variable x_(i) in the obtained polynomial on a bounded integerdomain, herein it introduces d^(x) ^(i) binary variables y₁ ^(x) ^(i) ,y₂ ^(x) ^(i) , . . . , y_(d) _(x) _(i) ^(x) ^(i) .

It is appreciated that the coefficients of the polynomial in severalbinary variables can be computed in various embodiments.

In one embodiment the computation of the coefficient of the polynomialin several binary variables may be performed according to the methoddisclosed in the documentation of the SymPy Python library for symbolicmathematics available online athttp://docs.sympy.org/latest/modules/polys/internals.html in conjunctionto the relations of type y^(m)=y for all binary variables.

In a particular case that the obtained polynomial on a bounded integerdomain is linear, the resulting polynomial in binary variables is alsolinear and the coefficient of each variable y_(k) ^(x) ^(i) isq_(i)c_(k) ^(x) ^(i) for i=1, . . . , n and k=1, . . . , d^(x) ^(i) .

In a particular case that the obtained polynomial on a bounded integerdomain is of degree two, then the equivalent polynomial in binaryvariables is of degree two as well; the coefficients of variable y_(k)^(x) ^(i) is q_(i)c_(k) ^(x) ^(i) +Q_(ii)(c_(k) ^(x) ^(i) )² for i=1, .. . , n, and k=1, . . . , d^(x) ^(i) ; the coefficients corresponding toy_(k) ^(x) ^(i) y_(l) ^(x) ^(i) is Q_(ii)c_(k) ^(x) ^(i) c_(l) ^(x) ^(i)for i=1, . . . n, k, l=1, . . . , d^(x) ^(i) , and k≠1; and thecoefficients corresponding to y_(k) ^(x) ^(i) y_(l) ^(x) ^(j) isQ_(ij)c_(k) ^(x) ^(i) c_(l) ^(x) ^(j) for i,j=1, . . . , n, i≠j, k=1, .. . , d^(x) ^(i) , and l=1, . . . , d^(x) ^(i) .

Degree Reduction of the Polynomial in Several Binary Variables

In some embodiments, the methods, systems, and media described hereininclude a series of steps for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding. In some embodiments, aprocessing step is shown to be providing a degree reduced form of apolynomial in several binary variables. Referring back to FIG. 8 andaccording to processing step 810, it provides a polynomial of degree atmost two in several binary variables equivalent to the providedpolynomial in several binary variables.

It is appreciated that the degree reduction of a polynomial in severalbinary variables can be done in various embodiments. In one embodimentthe degree reduction of a polynomial in several binary variables is doneby the method described by H. Ishikawa, “Transformation of GeneralBinary MRF Minimization to the First-Order Case,” in IEEE Transactionson Pattern Analysis and Machine Intelligence, vol. 33, no. 6, pp.1234-1249, June 2011]. In another embodiment, the degree reduction of apolynomial in several binary variables is done by the method describedin by Martin Anthony, Endre Boros, Yves Crama, and Aritanan Gruber,2016. Quadratization of symmetric pseudo-Boolean functions, DiscreteAppl. Math. 203, C (April 2016), 1-12.(DOI=http://dx.doi.org/10.1016/j.dam.2016.01.001).

Assigning Variables to Qubits

In some embodiments, the methods, systems, and media described hereininclude a series of steps for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding. In some embodiments, aprocessing step is shown to be providing an assignment of binaryvariables of the polynomial of degree at most two equivalent to theprovided polynomial on bounded integer domain to qubits. Referring backto FIG. 8 and according to processing step 812, it provides anassignment of the binary variables of the polynomial of degree at mosttwo equivalent to the provided polynomial on bounded integer domain toqubits. In one embodiment the assignment of binary variables to qubitsis according to a minor embedding algorithm from a source graph obtainedfrom the polynomial of degree at most two in several binary variablesequivalent to the provided polynomial on bounded integer domain to atarget graph obtained from the qubits and couplings of the pairs ofqubits in the system of superconducting qubits.

It is appreciated that a minor embedding from a source graph to a targetgraph may be performed according to various embodiments. In oneembodiment, the algorithm used is as disclosed in Jun Cai, Bill Macreadyand Aidan Roy, “A practical heuristic for finding graph minors”(arXiv:1406.2741) and in U.S. Patent Publication No. 2008/0218519 andU.S. Pat. No. 8,655,828 B2, each of which is entirely incorporatedherein by reference.

Setting Local Field Biases and Couplers Strengths

In some embodiments, the methods, systems, and media described hereininclude a series of steps for setting a system of superconducting qubitshaving a Hamiltonian representative of a polynomial on a bounded integerdomain via bounded-coefficient encoding. In some embodiments, aprocessing step is shown to be setting local field biases and couplingsstrengths. Referring back to FIG. 8 and according to processing step814, the local field biases and couplings strengths on the system ofsuperconducting qubits are tuned.

In the particular case that the obtained polynomial is linear, eachlogical variable is assigned a physical qubit and the local field biasof q_(i)c_(k) ^(x) ^(i) is assigned to the qubit corresponding tological variable y_(k) ^(x) ^(i) for i=1, . . . , n and k=1, . . . ,d^(x) ^(i) .

In the particular case where the obtained polynomial is of degree two ormore, the degree reduced polynomial in several binary variablesequivalent to the provided polynomial is quadratic, and the tuning oflocal field biases and coupling strength may be carried according tovarious embodiments. In one embodiment wherein the system ofsuperconducting qubits is fully connected, each logical variable isassigned a physical qubit. In this case, the local field of qubitcorresponding to variable y is set as the value of the coefficient of yin the polynomial of degree at most two in several binary variables. Thecoupling strength of the pair of qubits corresponding to variables y andy′ is set as the value of the coefficient of yy′ in the polynomial ofdegree at most two in several binary variables.

The following example, illustrates how the method disclosed in thisapplication may be used to recast a mixed-integer polynomiallyconstrained polynomial programming problem to a binary polynomiallyconstrained polynomial programming problem. Consider the optimizationproblemmin(x ₁ +x ₃)² +x ₂,subject to x ₁+(x ₂)³≤9,x ₁ ,x ₂∈

₊,x ₃∈{0,1}.The above problem is a mixed-integer polynomially constrained polynomialprogramming problem in which all the polynomials are of degree at mostthree. According to the constraint an upper bound for the integervariable x₁ is 9 and an upper bound for the integer variable x₂ is 2.

Suppose one wants to convert this problem to an equivalent binarypolynomially constrained polynomial programming with an integer encodingthat has coefficient of at most three. The bounded-coefficient encodingfor x₁ is c₁ ^(x) ¹ =1, c₂ ^(x) ¹ =2, c₃ ^(x) ¹ =3, c₄ ^(x) ¹ =3 and thebounded-coefficient encoding for x₂ is c₁ ^(x) ² =1, c₂ ^(x) ² =1. Theformal presentations for x₁ and x₂ arex ₁ =y ₁ ^(x) ¹ +2y ₂ ^(x) ¹ +3y ₃ ^(x) ¹ +3y ₄ ^(x) ¹ ,x ₂ =y ₁ ^(x) ² +y ₂ ^(x) ² .

Substituting the above linear functions for x₁ and x₂ in the mixedinteger polynomially constrained polynomial programming problem it willattain the following equivalent binary polynomially constrainedpolynomial programming problem:min(y ₁ ^(x) ¹ +2y ₂ ^(x) ¹ +3y ₃ ^(x) ¹ +3y ₄ ^(x) ¹ +x ₃)² +y ₁ ^(x) ²+y ₁ ^(x) ² ,subject to y ₁ ^(x) ¹ +2y ₂ ^(x) ¹ +3y ₃ ^(x) ¹ +3y ₄ ^(x) ¹ +(y ₁ ^(x)² +y ₂ ^(x) ² )³≤9,x ₃ ,y ₁ ^(x) ¹ ,y ₂ ^(x) ¹ ,y ₃ ^(x) ¹ ,y ₄ ^(x) ¹ ,y ₁ ^(x) ² ,y ₂^(x) ² ∈{0,1}.

If required, it can rule out degenerate solutions by adding the systemof non-degeneracy constraints provided by the method disclosed in thisapplication, to the derived binary polynomially constrained polynomialprogramming problem as mentioned above. For the presented example, thefinal binary polynomially constrained polynomial programming problem is:min(y ₁ ^(x) ¹ +2y ₂ ^(x) ¹ +3y ₃ ^(x) ¹ +y ₄ ^(x) ¹ +x ₃)² +y ₁ ^(x) ²+y ₂ ^(x) ² ,subject to y ₁ ^(x) ¹ +2y ₂ ^(x) ¹ +3y ₃ ^(x) ¹ +3y ₄ ^(x) ¹ +(y ₁ ^(x)² +y ₂ ^(x) ² )³≤9,y ₁ ^(x) ¹ +y ₂ ^(x) ¹ ≥y ₃ ^(x) ¹ ,y ₃ ^(x) ¹ ≥y ₄ ^(x) ¹ ,y ₁ ^(x) ² ≥y ₂ ^(x) ² ,x ₃ ,y ₁ ^(x) ¹ ,y ₂ ^(x) ¹ ,y ₃ ^(x) ¹ ,y ₄ ^(x) ¹ ,y ₁ ^(x) ² ,y ₂^(x) ² ∈{0,1}.In this particular cases the first constraint of the above problem is ofdegree three and in the form ofy ₁ ^(x) ¹ +2y ₂ ^(x) ¹ +3y ₃ ^(x) ¹ +3y ₄ ^(x) ¹ +(y ₁ ^(x) ² )³+3(y ₁^(x) ² )²(y ₂ ^(x) ² )+3(y ₁ ^(x) ² )(y ₂ ^(x) ² )²+(y ₂ ^(x) ² )³≤9,which can be equivalently represented as the degree reduced form ofy ₁ ^(x) ¹ +2y ₂ ^(x) ¹ +3y ₃ ^(x) ¹ +3y ₄ ^(x) ¹ +(y ₁ ^(x) ² )+6(y ₁^(x) ² )(y ₂ ^(x) ² )+(y ₂ ^(x) ² )≤9.Digital Processing Device

In some embodiments, the quantum-ready software development kit (SDK)described herein include a digital processing device, or use of thesame. In further embodiments, the digital processing device includes oneor more hardware central processing units (CPU) that carry out thedevice's functions. In still further embodiments, the digital processingdevice further comprises an operating system configured to performexecutable instructions. In some embodiments, the digital processingdevice is optionally connected a computer network. In furtherembodiments, the digital processing device is optionally connected tothe Internet such that it accesses the World Wide Web. In still furtherembodiments, the digital processing device is optionally connected to acloud computing infrastructure. In other embodiments, the digitalprocessing device is optionally connected to an intranet. In otherembodiments, the digital processing device is optionally connected to adata storage device.

In accordance with the description herein, suitable digital processingdevices include, by way of non-limiting examples, server computers,desktop computers, laptop computers, notebook computers, sub-notebookcomputers, netbook computers, netpad computers, set-top computers, mediastreaming devices, handheld computers, Internet appliances, mobilesmartphones, tablet computers, personal digital assistants, video gameconsoles, and vehicles. Those of skill in the art will recognize thatmany smartphones are suitable for use in the system described herein.Those of skill in the art will also recognize that select televisions,video players, and digital music players with optional computer networkconnectivity are suitable for use in the system described herein.Suitable tablet computers include those with booklet, slate, andconvertible configurations, known to those of skill in the art.

In some embodiments, the digital processing device includes an operatingsystem configured to perform executable instructions. The operatingsystem is, for example, software, including programs and data, whichmanages the device's hardware and provides services for execution ofapplications. Those of skill in the art will recognize that suitableserver operating systems include, by way of non-limiting examples,FreeBSD, OpenBSD, NetBSD®, Linux, Apple® Mac OS X Server®, Oracle®Solaris®, Windows Server®, and Novell® NetWare®. Those of skill in theart will recognize that suitable personal computer operating systemsinclude, by way of non-limiting examples, Microsoft® Windows®, Apple®Mac OS X®, UNIX®, and UNIX-like operating systems such as GNU/Linux®. Insome embodiments, the operating system is provided by cloud computing.Those of skill in the art will also recognize that suitable mobile smartphone operating systems include, by way of non-limiting examples, Nokia®Symbian® OS, Apple® iOS®, Research In Motion® BlackBerry OS®, Google®Android®, Microsoft® Windows Phone® OS, Microsoft® Windows Mobile® OS,Linux, and Palm® WebOS®. Those of skill in the art will also recognizethat suitable media streaming device operating systems include, by wayof non-limiting examples, Apple TV®, Roku®, Boxee®, Google TV®, GoogleChromecast®, Amazon Fire®, and Samsung® HomeSync®. Those of skill in theart will also recognize that suitable video game console operatingsystems include, by way of non-limiting examples, Sony® PS3®, Sony®PS4®, Microsoft® Xbox 360®, Microsoft Xbox One, Nintendo® Wii®,Nintendo® Wii U®, and Ouya®.

In some embodiments, the device includes a storage and/or memory device.The storage and/or memory device is one or more physical apparatusesused to store data or programs on a temporary or permanent basis. Insome embodiments, the device is volatile memory and requires power tomaintain stored information. In some embodiments, the device isnon-volatile memory and retains stored information when the digitalprocessing device is not powered. In further embodiments, thenon-volatile memory comprises flash memory. In some embodiments, thenon-volatile memory comprises dynamic random-access memory (DRAM). Insome embodiments, the non-volatile memory comprises ferroelectric randomaccess memory (FRAM). In some embodiments, the non-volatile memorycomprises phase-change random access memory (PRAM). In otherembodiments, the device is a storage device including, by way ofnon-limiting examples, CD-ROMs, DVDs, flash memory devices, magneticdisk drives, magnetic tapes drives, optical disk drives, and cloudcomputing based storage. In further embodiments, the storage and/ormemory device is a combination of devices such as those disclosedherein.

In some embodiments, the digital processing device includes a display tosend visual information to a user. In some embodiments, the display is acathode ray tube (CRT). In some embodiments, the display is a liquidcrystal display (LCD). In further embodiments, the display is a thinfilm transistor liquid crystal display (TFT-LCD). In some embodiments,the display is an organic light emitting diode (OLED) display. Invarious further embodiments, on OLED display is a passive-matrix OLED(PMOLED) or active-matrix OLED (AMOLED) display. In some embodiments,the display is a plasma display. In other embodiments, the display is avideo projector. In still further embodiments, the display is acombination of devices such as those disclosed herein.

In some embodiments, the digital processing device includes an inputdevice to receive information from a user. In some embodiments, theinput device is a keyboard. In some embodiments, the input device is apointing device including, by way of non-limiting examples, a mouse,trackball, track pad, joystick, game controller, or stylus. In someembodiments, the input device is a touch screen or a multi-touch screen.In other embodiments, the input device is a microphone to capture voiceor other sound input. In other embodiments, the input device is a videocamera or other sensor to capture motion or visual input. In furtherembodiments, the input device is a Kinect, Leap Motion, or the like. Instill further embodiments, the input device is a combination of devicessuch as those disclosed herein.

Non-Transitory Computer Readable Storage Medium

In some embodiments, the quantum-ready software development kit (SDK)disclosed herein include one or more non-transitory computer readablestorage media encoded with a program including instructions executableby the operating system of an optionally networked digital processingdevice. In further embodiments, a computer readable storage medium is atangible component of a digital processing device. In still furtherembodiments, a computer readable storage medium is optionally removablefrom a digital processing device. In some embodiments, a computerreadable storage medium includes, by way of non-limiting examples,CD-ROMs, DVDs, flash memory devices, solid state memory, magnetic diskdrives, magnetic tape drives, optical disk drives, cloud computingsystems and services, and the like. In some cases, the program andinstructions are permanently, substantially permanently,semi-permanently, or non-transitorily encoded on the media.

Computer Program

In some embodiments, the quantum-ready software development kit (SDK)disclosed herein include at least one computer program, or use of thesame. A computer program includes a sequence of instructions, executablein the digital processing device's CPU, written to perform a specifiedtask. Computer readable instructions may be implemented as programmodules, such as functions, objects, Application Programming Interfaces(APIs), data structures, and the like, that perform particular tasks orimplement particular abstract data types. In light of the disclosureprovided herein, those of skill in the art will recognize that acomputer program may be written in various versions of variouslanguages.

The functionality of the computer readable instructions may be combinedor distributed as desired in various environments. In some embodiments,a computer program comprises one sequence of instructions. In someembodiments, a computer program comprises a plurality of sequences ofinstructions. In some embodiments, a computer program is provided fromone location. In other embodiments, a computer program is provided froma plurality of locations. In various embodiments, a computer programincludes one or more software modules. In various embodiments, acomputer program includes, in part or in whole, one or more webapplications, one or more mobile applications, one or more standaloneapplications, one or more web browser plug-ins, extensions, add-ins, oradd-ons, or combinations thereof.

Web Application

In some embodiments, a computer program includes a web application. Inlight of the disclosure provided herein, those of skill in the art willrecognize that a web application, in various embodiments, utilizes oneor more software frameworks and one or more database systems. In someembodiments, a web application is created upon a software framework suchas Microsoft® .NET or Ruby on Rails (RoR). In some embodiments, a webapplication utilizes one or more database systems including, by way ofnon-limiting examples, relational, non-relational, object oriented,associative, and XML database systems. In further embodiments, suitablerelational database systems include, by way of non-limiting examples,Microsoft® SQL Server, mySQL™, and Oracle®. Those of skill in the artwill also recognize that a web application, in various embodiments, iswritten in one or more versions of one or more languages. A webapplication may be written in one or more markup languages, presentationdefinition languages, client-side scripting languages, server-sidecoding languages, database query languages, or combinations thereof. Insome embodiments, a web application is written to some extent in amarkup language such as Hypertext Markup Language (HTML), ExtensibleHypertext Markup Language (XHTML), or eXtensible Markup Language (XML).In some embodiments, a web application is written to some extent in apresentation definition language such as Cascading Style Sheets (CSS).In some embodiments, a web application is written to some extent in aclient-side scripting language such as Asynchronous Javascript and XML(AJAX), Flash® Actionscript, Javascript, or Silverlight®. In someembodiments, a web application is written to some extent in aserver-side coding language such as Active Server Pages (ASP),ColdFusion®, Perl, Java™, JavaServer Pages (JSP), Hypertext Preprocessor(PHP), Python™, Ruby, Tcl, Smalltalk, WebDNA®, or Groovy. In someembodiments, a web application is written to some extent in a databasequery language such as Structured Query Language (SQL). In someembodiments, a web application integrates enterprise server productssuch as IBM® Lotus Domino®. In some embodiments, a web applicationincludes a media player element. In various further embodiments, a mediaplayer element utilizes one or more of many suitable multimediatechnologies including, by way of non-limiting examples, Adobe® Flash®,HTML 5, Apple® QuickTime®, Microsoft® Silverlight®, Java™, and Unity®.

Mobile Application

In some embodiments, a computer program includes a mobile applicationprovided to a mobile digital processing device. In some embodiments, themobile application is provided to a mobile digital processing device atthe time it is manufactured. In other embodiments, the mobileapplication is provided to a mobile digital processing device via thecomputer network described herein.

In view of the disclosure provided herein, a mobile application may becreated by techniques known to those of skill in the art using hardware,languages, and development environments known to the art. Those of skillin the art will recognize that mobile applications are written inseveral languages. Suitable programming languages include, by way ofnon-limiting examples, C, C++, C#, Objective-C, Java™, Javascript,Pascal, Object Pascal, Python™, Ruby, VB.NET, WML, and XHTML/HTML withor without CSS, or combinations thereof.

Suitable mobile application development environments are available fromseveral sources. Commercially available development environmentsinclude, by way of non-limiting examples, AirplaySDK, alcheMo,Appcelerator®, Celsius, Bedrock, Flash Lite, .NET Compact Framework,Rhomobile, and WorkLight Mobile Platform. Other development environmentsare available without cost including, by way of non-limiting examples,Lazarus, MobiFlex, MoSync, and Phonegap. Also, mobile devicemanufacturers distribute software developer kits including, by way ofnon-limiting examples, iPhone and iPad (iOS) SDK, Android™ SDK,BlackBerry® SDK, BREW SDK, Palm® OS SDK, Symbian SDK, webOS SDK, andWindows® Mobile SDK.

Those of skill in the art will recognize that several commercial forumsare available for distribution of mobile applications including, by wayof non-limiting examples, Apple® App Store, Android™ Market, BlackBerry®App World, App Store for Palm devices, App Catalog for webOS, Windows®Marketplace for Mobile, Ovi Store for Nokia® devices, Samsung® Apps, andNintendo® DSi Shop.

Standalone Application

In some embodiments, a computer program includes a standaloneapplication, which is a program that is run as an independent computerprocess, not an add-on to an existing process, e.g., not a plug-in.Those of skill in the art will recognize that standalone applicationsare often compiled. A compiler is a computer program(s) that transformssource code written in a programming language into binary object codesuch as assembly language or machine code. Suitable compiled programminglanguages include, by way of non-limiting examples, C, C++, Objective-C,COBOL, Delphi, Eiffel, Java™, Lisp, Python™, Visual Basic, and VB .NET,or combinations thereof. Compilation is often performed, at least inpart, to create an executable program. In some embodiments, a computerprogram includes one or more executable complied applications.

Web Browser Plug-In

In some embodiments, the computer program includes a web browserplug-in. In computing, a plug-in is one or more software components thatadd specific functionality to a larger software application. Makers ofsoftware applications support plug-ins to enable third-party developersto create abilities which extend an application, to support easilyadding new features, and to reduce the size of an application. Whensupported, plug-ins enable customizing the functionality of a softwareapplication. For example, plug-ins are commonly used in web browsers toplay video, generate interactivity, scan for viruses, and displayparticular file types. Those of skill in the art will be familiar withseveral web browser plug-ins including, Adobe® Flash® Player, Microsoft®Silverlight®, and Apple® QuickTime®. In some embodiments, the toolbarcomprises one or more web browser extensions, add-ins, or add-ons. Insome embodiments, the toolbar comprises one or more explorer bars, toolbands, or desk bands.

In view of the disclosure provided herein, those of skill in the artwill recognize that several plug-in frameworks are available that enabledevelopment of plug-ins in various programming languages, including, byway of non-limiting examples, C++, Delphi, Java™, PHP, Python™, and VB.NET, or combinations thereof.

Web browsers (also called Internet browsers) are software applications,designed for use with network-connected digital processing devices, forretrieving, presenting, and traversing information resources on theWorld Wide Web. Suitable web browsers include, by way of non-limitingexamples, Microsoft® Internet Explorer®, Mozilla® Firefox®, Google®Chrome, Apple® Safari®, Opera Software® Opera®, and KDE Konqueror. Insome embodiments, the web browser is a mobile web browser. Mobile webbrowsers (also called mircrobrowsers, mini-browsers, and wirelessbrowsers) are designed for use on mobile digital processing devicesincluding, by way of non-limiting examples, handheld computers, tabletcomputers, netbook computers, subnotebook computers, smartphones, musicplayers, personal digital assistants (PDAs), and handheld video gamesystems. Suitable mobile web browsers include, by way of non-limitingexamples, Google® Android® browser, RIM BlackBerry® Browser, Apple®Safari®, Palm® Blazer, Palm® WebOS® Browser, Mozilla® Firefox® formobile, Microsoft® Internet Explorer® Mobile, Amazon® Kindle® Basic Web,Nokia® Browser, Opera Software® Opera® Mobile, and Sony® PSP™ browser.

Software Modules

In some embodiments, the quantum-ready software development kit (SDK)disclosed herein include software, server, and/or database modules, oruse of the same. In view of the disclosure provided herein, softwaremodules may be created by techniques known to those of skill in the artusing machines, software, and languages known to the art. The softwaremodules disclosed herein are implemented in a multitude of ways. Invarious embodiments, a software module comprises a file, a section ofcode, a programming object, a programming structure, or combinationsthereof. In further various embodiments, a software module comprises aplurality of files, a plurality of sections of code, a plurality ofprogramming objects, a plurality of programming structures, orcombinations thereof. In various embodiments, the one or more softwaremodules comprise, by way of non-limiting examples, a web application, amobile application, and a standalone application. In some embodiments,software modules are in one computer program or application. In otherembodiments, software modules are in more than one computer program orapplication. In some embodiments, software modules are hosted on onemachine. In other embodiments, software modules are hosted on more thanone machine. In further embodiments, software modules are hosted oncloud computing platforms. In some embodiments, software modules arehosted on one or more machines in one location. In other embodiments,software modules are hosted on one or more machines in more than onelocation.

Databases

In some embodiments, the quantum-ready software development kit (SDK)disclosed herein include one or more databases, or use of the same. Inview of the disclosure provided herein, those of skill in the art willrecognize that many databases are suitable for storage and retrieval ofapplication information. In various embodiments, suitable databasesinclude, by way of non-limiting examples, relational databases,non-relational databases, object oriented databases, object databases,entity-relationship model databases, associative databases, and XMLdatabases. In some embodiments, a database is internet-based. In furtherembodiments, a database is web-based. In still further embodiments, adatabase is cloud computing-based. In other embodiments, a database isbased on one or more local computer storage devices.

While preferred embodiments of the present invention have been shown anddescribed herein, it will be obvious to those skilled in the art thatsuch embodiments are provided by way of example only. It is not intendedthat the invention be limited by the specific examples provided withinthe specification. While the invention has been described with referenceto the aforementioned specification, the descriptions and illustrationsof the embodiments herein are not meant to be construed in a limitingsense. Numerous variations, changes, and substitutions will now occur tothose skilled in the art without departing from the invention.Furthermore, it shall be understood that all aspects of the inventionare not limited to the specific depictions, configurations or relativeproportions set forth herein which depend upon a variety of conditionsand variables. It should be understood that various alternatives to theembodiments of the invention described herein may be employed inpracticing the invention. It is therefore contemplated that theinvention shall also cover any such alternatives, modifications,variations or equivalents. It is intended that the following claimsdefine the scope of the invention and that methods and structures withinthe scope of these claims and their equivalents be covered thereby.

What is claimed is:
 1. A method for using a digital computer to generateand direct a computational task to a quantum computing resourcecomprising at least one quantum computer over a network, wherein saiddigital computer comprises at least one computer processor and at leastone computer memory, said method comprising: (a) retrieving aprogramming problem from said computer memory of said digital computer;(b) using said at least one computer processor of said digital computerto generate an equivalent of said programming problem; (c) generating arequest comprising said equivalent of said programming problem generatedby said at least one computer processor of said digital computer; and(d) directing said request from said digital computer to said quantumcomputing resource over said network, wherein said equivalent of saidprogramming problem is usable by said at least one quantum computer ofsaid quantum computing resource to solve said programming problem. 2.The method of claim 1, wherein said request is directed from saiddigital computer to said quantum computing resource through acloud-based interface.
 3. The method of claim 1, wherein said network isa local network.
 4. The method of claim 1, wherein said at least onequantum computer performs one or more quantum algorithms to saidprogramming problem.
 5. The method of claim 1, wherein said request isgenerated using an application programming interface (API).
 6. Themethod of claim 1, wherein (a) comprises obtaining (i) a polynomial on abounded integer domain and (ii) integer encoding parameters, and wherein(b) comprises computing a bounded-coefficient encoding using saidinteger encoding parameters.
 7. The method of claim 6, wherein (b)further comprises using said at least one computer processor totransform each integer variable of said polynomial to a linear functionof binary variables using said bounded-coefficient encoding.
 8. Themethod of claim 7, further comprising providing constraints on saidbinary variables to avoid degeneracy in said bounded-coefficientencoding, if required by a user.
 9. The method of claim 7, furthercomprising substituting each integer variable of said polynomial with anequivalent binary representation, and using said at least one computerprocessor to compute coefficients of said equivalent binaryrepresentation of said polynomial on said bounded integer domain. 10.The method of claim 9, further comprising performing a degree reductionon said equivalent binary representation of said polynomial on saidbounded integer domain to generate an equivalent polynomial.
 11. Themethod of claim 10, wherein said equivalent polynomial is of a degree ofat most two in binary variables.
 12. The method of claim 11, furthercomprising setting local field biases and coupling strengths on said atleast one quantum computer using said coefficients of said equivalentpolynomial of said degree of at most two in binary variables to generatesaid equivalent of said programming problem.
 13. The method of claim 12,wherein said equivalent of said programming problem comprises aHamiltonian representative of said polynomial on said bounded integerdomain.
 14. The method of claim 13, wherein said Hamiltonian is usableby said at least one quantum computer to solve said programming problem.15. A system comprising a digital computer for generating and directinga computational task to a quantum computing resource comprising at leastone quantum computer over a network, wherein said digital computercomprises at least one computer processor and at least one computermemory, wherein said at least one computer processor is programmed to:(a) retrieve a programming problem from said computer memory of saiddigital computer; (b) use said at least one computer processor of saiddigital computer to generate an equivalent of said programming problem;(c) generate a request comprising said equivalent of said programmingproblem generated by said at least one computer processor of saiddigital computer; and (d) direct said request from said digital computerto said quantum computing resource over said network, wherein saidequivalent of said programming problem is usable by said at least onequantum computer of said quantum computing resource to solve saidprogramming problem.
 16. The system of claim 15, wherein said at leastone computer processor is programmed to direct said request from saiddigital computer to said quantum computing resource through acloud-based interface.
 17. The system of claim 15, wherein said networkis a local network.
 18. The system of claim 15, wherein said at leastone computer processor is programmed to obtain (i) a polynomial on abounded integer domain and (ii) integer encoding parameters, and computea bounded-coefficient encoding using said integer encoding parameters.19. The system of claim 18, wherein said at least one computer processoris programmed to (i) transform each integer variable of said polynomialto a linear function of binary variables using said bounded-coefficientencoding, and (ii) substitute each integer variable of said polynomialwith an equivalent binary representation, and compute coefficients of anequivalent binary representation of said polynomial on said boundedinteger domain.
 20. The system of claim 19, wherein said at least onecomputer processor is programmed to perform a degree reduction on saidequivalent binary representation of said polynomial on said boundedinteger domain to generate an equivalent polynomial, wherein saidequivalent polynomial is of a degree of at most two in binary variables.21. The system of claim 20, wherein said at least one computer processoris programmed to set local field biases and coupling strengths on saidat least one quantum computer using said coefficients of said equivalentpolynomial of said degree of at most two in binary variables to generatesaid equivalent of said programming problem.
 22. The system of claim 21,wherein said equivalent of said programming problem comprises aHamiltonian representative of said polynomial on said bounded integerdomain.
 23. The system of claim 22, wherein said Hamiltonian is usableby said at least one quantum computer to solve said programming problem.